2005
2005
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1.I.1C
2005 commentConvert the following expressions from suffix notation (assuming the summation convention in three dimensions) into standard notation using vectors and/or matrices, where possible, identifying the one expression that is incorrectly formed:
(i) ,
(ii) ,
(iii) ,
(iv) ,
(v) .
Write the vector triple product in suffix notation and derive an equivalent expression that utilises scalar products. Express the result both in suffix notation and in standard vector notation. Hence or otherwise determine when and are orthogonal and .
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1.I.2B
2005 commentLet be a unit vector. Consider the operation
Write this in matrix form, i.e., find a matrix such that for all , and compute the eigenvalues of . In the case when , compute and its eigenvalues and eigenvectors.
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1.II.5C
2005 commentGive the real and imaginary parts of each of the following functions of , with real, (i) , (ii) , (iii) , (iv) , (v) ,
where is the complex conjugate of .
An ant lives in the complex region given by . Food is found at such that
Drink is found at such that
Identify the places within where the ant will find the food or drink.
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1.II.6B
2005 commentLet be a real matrix. Define the rank of . Describe the space of solutions of the equation
organizing your discussion with reference to the rank of .
Write down the equation of the tangent plane at on the sphere and the equation of a general line in passing through the origin .
Express the problem of finding points on the intersection of the tangent plane and the line in the form . Find, and give geometrical interpretations of, the solutions.
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1.II.7A
2005 commentConsider two vectors and in . Show that a may be written as the sum of two vectors: one parallel (or anti-parallel) to and the other perpendicular to . By setting the former equal to , where is a unit vector along , show that
Explain why this is a sensible definition of the angle between and .
Consider the vertices of a cube of side 2 in , centered on the origin. Each vertex is joined by a straight line through the origin to another vertex: the lines are the diagonals of the cube. Show that no two diagonals can be perpendicular if is odd.
For , what is the greatest number of mutually perpendicular diagonals? List all the possible angles between the diagonals.
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1.II.8A
2005 commentGiven a non-zero vector , any symmetric matrix can be expressed as
for some numbers and , some vector and a symmetric matrix , where
and the summation convention is implicit.
Show that the above statement is true by finding and explicitly in terms of and , or otherwise. Explain why and together provide a space of the correct dimension to parameterise an arbitrary symmetric matrix .
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3.I.1D
2005 commentLet be a real symmetric matrix with eigenvalues . Consider the surface in given by
Find the minimum distance between the origin and . How many points on realize this minimum distance? Justify your answer.
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3.I.2D
2005 commentDefine what it means for a group to be cyclic. If is a prime number, show that a finite group of order must be cyclic. Find all homomorphisms , where denotes the cyclic group of order . [You may use Lagrange's theorem.]
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3.II.7D
2005 commentLet be a group and let for all . Show that is a normal subgroup of
Let be the set of all real matrices of the form
with . Show that is a subgroup of the group of invertible real matrices under multiplication.
Find and show that is isomorphic to with vector addition.
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3.II.8D
2005 commentLet be a real matrix such that , and , where is the transpose of and is the identity.
Show that the set of vectors for which forms a 1-dimensional subspace.
Consider the plane through the origin which is orthogonal to . Show that maps to itself and induces a rotation of by angle , where . Show that is a reflection in if and only if has trace 1 . [You may use the fact that for any invertible matrix B.]
Prove that .
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3.II.5D
2005 commentDefine the notion of an action of a group on a set . Assuming that is finite, state and prove the Orbit-Stabilizer Theorem.
Let be a finite group and the set of its subgroups. Show that defines an action of on . If is a subgroup of , show that the orbit of has at most elements.
Suppose is a subgroup of and . Show that there is an element of which does not belong to any subgroup of the form for .
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3.II.6D
2005 commentLet be the group of Möbius transformations of and let be the group of all complex matrices with determinant 1 .
Show that the map given by
is a surjective homomorphism. Find its kernel.
Show that every not equal to the identity is conjugate to a Möbius map where either with , or . [You may use results about matrices in , provided they are clearly stated.]
Show that if , then is the identity, or has one, or two, fixed points. Also show that if has only one fixed point then as for any
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1.I.3F
2005 commentDefine the supremum or least upper bound of a non-empty set of real numbers.
Let denote a non-empty set of real numbers which has a supremum but no maximum. Show that for every there are infinitely many elements of contained in the open interval
Give an example of a non-empty set of real numbers which has a supremum and maximum and for which the above conclusion does not hold.
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1.I.4D
2005 commentLet be a power series in the complex plane with radius of convergence . Show that is unbounded in for any with . State clearly any results on absolute convergence that are used.
For every , show that there exists a power series with radius of convergence .
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1.II.9F
2005 commentExamine each of the following series and determine whether or not they converge.
Give reasons in each case.
(iii)
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1.II.10D
2005 commentExplain what it means for a bounded function to be Riemann integrable.
Let be a strictly decreasing continuous function. Show that for each , there exists a unique point such that
Find if .
Suppose now that is differentiable and for all . Prove that is differentiable at all and for all , stating clearly any results on the inverse of you use.
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1.II.11E
2005 commentProve that if is a continuous function on the interval with then for some .
Let be a continuous function on satisfying . By considering the function on , show that for some . Show, more generally, that for any positive integer there exists a point for which .
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1.II.12E
2005 commentState and prove Rolle's Theorem.
Prove that if the real polynomial of degree has all its roots real (though not necessarily distinct), then so does its derivative . Give an example of a cubic polynomial for which the converse fails.
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2.I.1B
2005 commentSolve the equation
with , by use of an integrating factor or otherwise. Find .
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2.I.2B
2005 commentObtain the general solution of
by using the indicial equation.
Introduce as a new independent variable and find an equivalent second order differential equation with constant coefficients. Determine the general solution of this new equation, and show that it is equivalent to the general solution of found previously.
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2.II.5B
2005 commentFind two linearly independent solutions of the difference equation
for all values of . What happens when ? Find two linearly independent solutions in this case.
Find which satisfy the initial conditions
for and for . For every , show that as .
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2005
commentFind all power series solutions of the form to the equation
for a real constant.
Impose the condition and determine those values of for which your power series gives polynomial solutions (i.e., for sufficiently large). Give the values of for which the corresponding polynomials have degree less than 6 , and compute these polynomials.
Hence, or otherwise, find a polynomial solution of
satisfying .
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2.II.7B
2005 commentThe Cartesian coordinates of a point moving in are governed by the system
Transform this system of equations to polar coordinates and hence find all periodic solutions (i.e., closed trajectories) which satisfy constant.
Discuss the large time behaviour of an arbitrary solution starting at initial point . Summarize the motion using a phase plane diagram, and comment on the nature of any critical points.
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2.II.8B
2005 commentDefine the Wronskian for two solutions of the equation
and obtain a differential equation which exhibits its dependence on . Explain the relevance of the Wronskian to the linear independence of and .
Consider the equation
and determine the dependence on of the Wronskian of two solutions and . Verify that is a solution of and use the Wronskian to obtain a second linearly independent solution.
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4.I.3C
2005 commentPlanetary Explorers Ltd. want to put a communications satellite of mass into geostationary orbit around the spherical planet Zog (i.e. with the satellite always above the same point on the surface of Zog). The mass of Zog is , the length of its day is and is the gravitational constant.
Write down the equations of motion for a general orbit of the satellite and determine the radius and speed of the geostationary orbit.
Describe briefly how the orbit is modified if the satellite is released at the correct radius and on the correct trajectory for a geostationary orbit, but with a little too much speed. Comment on how the satellite's speed varies around such an orbit.
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4.I.4C
2005 commentA car of mass travelling at speed on a smooth, horizontal road attempts an emergency stop. The car skids in a straight line with none of its wheels able to rotate.
Calculate the stopping distance and time on a dry road where the dry friction coefficient between the tyres and the road is .
At high speed on a wet road the grip of each of the four tyres changes from dry friction to a lubricated drag equal to for each tyre, where is the drag coefficient and the instantaneous speed of the car. However, the tyres regain their dry-weather grip when the speed falls below . Calculate the stopping distance and time under these conditions.
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4.II.9C
2005 commentA particle of mass and charge moving in a vacuum through a magnetic field and subject to no other forces obeys
where is the location of the particle.
For with constant , and using cylindrical polar coordinates , or otherwise, determine the motion of the particle in the plane if its initial speed is with . [Hint: Choose the origin so that and at .]
Due to a leak, a small amount of gas enters the system, causing the particle to experience a drag force , where . Write down the new governing equations and show that the speed of the particle decays exponentially. Sketch the path followed by the particle. [Hint: Consider the equations for the velocity in Cartesian coordinates; you need not apply any initial conditions.]
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4.II.10C
2005 commentA keen cyclist wishes to analyse her performance on training rollers. She decides that the key components are her bicycle's rear wheel and the roller on which the wheel sits. The wheel, of radius , has its mass entirely at its outer edge. The roller, which is driven by the wheel without any slippage, is a solid cylinder of radius and mass . The angular velocities of the wheel and roller are and , respectively.
Determine and , the moments of inertia of the wheel and roller, respectively. Find the ratio of the angular velocities of the wheel and roller. Show that the combined total kinetic energy of the wheel and roller is , where
is the effective combined moment of inertia of the wheel and roller.
Why should be used instead of just or in the equation connecting torque with angular acceleration? The cyclist believes the torque she can produce at the back wheel is where and are dimensional constants. Determine the angular velocity of the wheel, starting from rest, as a function of time.
In an attempt to make the ride more realistic, the cyclist adds a fan (of negligible mass) to the roller. The fan imposes a frictional torque on the roller, where is a dimensional constant. Determine the new maximum speed for the wheel.
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4.II.11C
2005 commentA puck of mass located at slides without friction under the influence of gravity on a surface of height . Show that the equations of motion can be approximated by
where is the gravitational acceleration and the small slope approximation is used.
Determine the motion of the puck when .
Sketch the surface
as a function of , where . Write down the equations of motion of the puck on this surface in polar coordinates under the assumption that the small slope approximation can be used. Show that , the angular momentum per unit mass about the origin, is conserved. Show also that the initial kinetic energy per unit mass of the puck is if the puck is released at radius with negligible radial velocity. Determine and sketch as a function of for this release condition. What condition relating and must be satisfied for the orbit to be bounded?
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4.II.12C
2005 commentIn an experiment a ball of mass is released from a height above a flat, horizontal plate. Assuming the gravitational acceleration is constant and the ball falls through a vacuum, find the speed of the ball on impact.
Determine the speed at which the ball rebounds if the coefficient of restitution for the collision is . What fraction of the impact energy is dissipated during the collision? Determine also the maximum height the ball reaches after the bounce, and the time between the and bounce. What is the total distance travelled by the ball before it comes to rest if ?
If the experiment is repeated in an atmosphere then the ball experiences a drag force , where is a dimensional constant and the instantaneous velocity of the ball. Write down and solve the modified equation for before the ball first hits the plate.
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4.I.1E
2005 commentFind the unique positive integer with , for which
Results used should be stated but need not be proved.
Solve the system of simultaneous congruences
Explain very briefly your reasoning.
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4.I.2E
2005 commentGive a combinatorial definition of the binomial coefficient for any non-negative integers .
Prove that for .
Prove the identities
and
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4.II.5E
2005 commentWhat does it mean for a set to be countable? Show that is countable, and is not countable.
Let be any set of non-trivial discs in a plane, any two discs being disjoint. Show that is countable.
Give an example of a set of non-trivial circles in a plane, any two circles being disjoint, which is not countable.
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4.II.6E
2005 commentLet be a relation on the set . What does it mean for to be an equivalence relation on ? Show that if is an equivalence relation on , the set of equivalence classes forms a partition of .
Let be a group, and let be a subgroup of . Define a relation on by if . Show that is an equivalence relation on , and that the equivalence classes are precisely the left cosets of in . Find a bijection from to any other coset . Deduce that if is finite then the order of divides the order of .
Let be an element of the finite group . The order of is the least positive integer for which , the identity of . If , then has a subgroup of order ; deduce that for all .
Let be a natural number. Show that the set of integers in which are prime to is a group under multiplication modulo . [You may use any properties of multiplication and divisibility of integers without proof, provided you state them clearly.]
Deduce that if is any integer prime to then , where is the Euler totient function.
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4.II
2005 commentState and prove the Principle of Inclusion and Exclusion.
Use the Principle to show that the Euler totient function satisfies
Deduce that if and are coprime integers, then , and more generally, that if is any divisor of then divides .
Show that if divides then for some non-negative integers .
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4.II.8E
2005 commentThe Fibonacci numbers are defined by the equations and for any positive integer . Show that the highest common factor is
Let be a natural number. Prove by induction on that for all positive integers ,
Deduce that divides for all positive integers . Deduce also that if then
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2.I.3F
2005 commentSuppose and is a positive real-valued random variable with probability density
for , where is a constant.
Find the constant and show that, if and ,
[You may assume the inequality for all .]
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2.I.4F
2005 commentDescribe the Poisson distribution characterised by parameter . Calculate the mean and variance of this distribution in terms of .
Show that the sum of independent random variables, each having the Poisson distribution with , has a Poisson distribution with .
Use the central limit theorem to prove that
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2.II
2005 commentGiven a real-valued random variable , we define by
Consider a second real-valued random variable , independent of . Show that
You gamble in a fair casino that offers you unlimited credit despite your initial wealth of 0 . At every game your wealth increases or decreases by with equal probability . Let denote your wealth after the game. For a fixed real number , compute defined by
Verify that the result is real-valued.
Show that for even,
for some constant , which you should determine. What is for odd?
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2.II.10F
2005 commentAlice and Bill fight a paint-ball duel. Nobody has been hit so far and they are both left with one shot. Being exhausted, they need to take a breath before firing their last shot. This takes seconds for Alice and seconds for Bill. Assume these times are exponential random variables with means and , respectively.
Find the distribution of the (random) time that passes by before the next shot is fired. What is its standard deviation? What is the probability that Alice fires the next shot?
Assume Alice has probability of hitting whenever she fires whereas Bill never misses his target. If the next shot is a hit, what is the probability that it was fired by Alice?
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2.II.11F
2005 commentLet be uniformly distributed on and define . Show that, conditionally on
the vector is uniformly distributed on the unit disc. Let denote the point in polar coordinates and find its probability density function for . Deduce that and are independent.
Introduce the new random variables
noting that under the above conditioning, are uniformly distributed on the unit disc. The pair may be viewed as a (random) point in with polar coordinates . Express as a function of and deduce its density. Find the joint density of . Hence deduce that and are independent normal random variables with zero mean and unit variance.
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2.II.12F
2005 commentLet be a ranking of the yearly rainfalls in Cambridge over the next years: assume is a random permutation of . Year is called a record year if for all (thus the first year is always a record year). Let if year is a record year and 0 otherwise.
Find the distribution of and show that are independent and calculate the mean and variance of the number of record years in the next years.
Find the probability that the second record year occurs at year . What is the expected number of years until the second record year occurs?
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3.I.3A
2005 commentLet and be time-dependent, continuously differentiable vector fields on satisfying
Show that for any bounded region ,
where is the boundary of .
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3.I.4A
2005 commentGiven a curve in , parameterised such that and with , define the tangent , the principal normal , the curvature and the binormal .
The torsion is defined by
Sketch a circular helix showing and at a chosen point. What is the sign of the torsion for your helix? Sketch a second helix with torsion of the opposite sign.
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3.II.9A
2005 commentLet be a bounded region of and be its boundary. Let be the unique solution to in , with on , where is a given function. Consider any smooth function also equal to on . Show, by using Green's first theorem or otherwise, that
[Hint: Set
Consider the partial differential equation
for , with initial condition in , and boundary condition on for all . Show that
with equality holding only when .
Show that remains true with the boundary condition
on , provided .
3/II/10A Vector Calculus
Write down Stokes' theorem for a vector field on .
Consider the bounded surface defined by
Sketch the surface and calculate the surface element . For the vector field
calculate directly.
Show using Stokes' theorem that may be rewritten as a line integral and verify this yields the same result.
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3.II.11A
2005 commentExplain, with justification, the significance of the eigenvalues of the Hessian in classifying the critical points of a function . In what circumstances are the eigenvalues inconclusive in establishing the character of a critical point?
Consider the function on ,
Find and classify all of its critical points, for all real . How do the locations of the critical points change as ?
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3.II.12A
2005 commentExpress the integral
in terms of the new variables , and . Hence show that
You may assume and are positive. [Hint: Remember to calculate the limits of the integral.]
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1.I.1C
2005 commentLet be an -dimensional vector space over , and let be a linear map. Define the minimal polynomial of . Prove that is invertible if and only if the constant term of the minimal polynomial of is non-zero.
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1.II.9C
2005 commentLet be a finite dimensional vector space over , and be the dual space of .
If is a subspace of , we define the subspace of by
Prove that . Deduce that, if is any real -matrix of rank , the equations
have linearly independent solutions in .
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2.I.1C
2005 commentLet be the set of all matrices of the form , where are in , and
Prove that is closed under multiplication and determine its dimension as a vector space over . Prove that
and deduce that each non-zero element of is invertible.
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2.II.10C
2005 comment(i) Let be an matrix with entries in C. Define the determinant of , the cofactor of each , and the adjugate matrix . Assuming the expansion of the determinant of a matrix in terms of its cofactors, prove that
where is the identity matrix.
(ii) Let
Show the eigenvalues of are , where , and determine the diagonal matrix to which is similar. For each eigenvalue, determine a non-zero eigenvector.
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3.II.10B
2005 commentLet be the vector space of functions such that the th derivative of is defined and continuous for every . Define linear maps by and . Show that
where in this question means and is the identity map on .
Now let be any real vector space with linear maps such that . Suppose that there is a nonzero element with . Let be the subspace of spanned by , and so on. Show that is in and give a formula for it. More generally, show that is in for each , and give a formula for it.
Show, using your formula or otherwise, that are linearly independent. (Or, equivalently: show that are linearly independent for every .)
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4.I.1B
2005 commentDefine what it means for an complex matrix to be unitary or Hermitian. Show that every eigenvalue of a Hermitian matrix is real. Show that every eigenvalue of a unitary matrix has absolute value 1 .
Show that two eigenvectors of a Hermitian matrix that correspond to different eigenvalues are orthogonal, using the standard inner product on .
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4.II.10B
2005 comment(i) Let be a finite-dimensional real vector space with an inner product. Let be a basis for . Prove by an explicit construction that there is an orthonormal basis for such that the span of is equal to the span of for every .
(ii) For any real number , consider the quadratic form
on . For which values of is nondegenerate? When is nondegenerate, compute its signature in terms of .
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1.II.10C
2005 commentLet be a group, and a subgroup of finite index. By considering an appropriate action of on the set of left cosets of , prove that always contains a normal subgroup of such that the index of in is finite and divides !, where is the index of in .
Now assume that is a finite group of order , where and are prime numbers with . Prove that the subgroup of generated by any element of order is necessarily normal.
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2.I.2C
2005 commentDefine an automorphism of a group , and the natural group law on the set of all automorphisms of . For each fixed in , put for all in . Prove that is an automorphism of , and that defines a homomorphism from into .
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2.II.11C
2005 commentLet be the abelian group generated by two elements , subject to the relation . Give a rigorous explanation of this statement by defining as an appropriate quotient of a free abelian group of rank 2. Prove that itself is not a free abelian group, and determine the exact structure of .
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3.I.1C
2005 commentDefine what is meant by two elements of a group being conjugate, and prove that this defines an equivalence relation on . If is finite, sketch the proof that the cardinality of each conjugacy class divides the order of .
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2005
comment(i) Define a primitive polynomial in , and prove that the product of two primitive polynomials is primitive. Deduce that is a unique factorization domain.
(ii) Prove that
is a field. Show, on the other hand, that
is an integral domain, but is not a field.
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4.I.2C
2005 commentState Eisenstein's irreducibility criterion. Let be an integer . Prove that is irreducible in if and only if is a prime number.
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4.II.11C
2005 commentLet be the ring of Gaussian integers , where , which you may assume to be a unique factorization domain. Prove that every prime element of divides precisely one positive prime number in . List, without proof, the prime elements of , up to associates.
Let be a prime number in . Prove that has cardinality . Prove that is not a field. If , show that is a field. If , decide whether is a field or not, justifying your answer.
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1.I
2005 commentLet be the map defined by
where . Describe briefly the image . Let denote the open subset of given by ; prove that the restriction defines a smooth parametrization of a certain open subset (which you should specify) of . Hence, or otherwise, prove that is a smooth embedded surface in .
[You may assume that the image under of any open set is open in .]
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2.II.12A
2005 commentLet be an open subset of equipped with a Riemannian metric. For a smooth curve, define what is meant by its length and energy. Prove that length , with equality if and only if has constant norm with respect to the metric.
Suppose now is the upper half plane model of the hyperbolic plane, and are points on the positive imaginary axis. Show that a smooth curve joining and represents an absolute minimum of the length of such curves if and only if , with a smooth monotonic real function.
Suppose that a smooth curve joining the above points and represents a stationary point for the energy under proper variations; deduce from an appropriate form of the Euler-Lagrange equations that must be of the above form, with constant.
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3.I.2A
2005 commentWrite down the Riemannian metric on the disc model of the hyperbolic plane. Given that the length minimizing curves passing through the origin correspond to diameters, show that the hyperbolic circle of radius centred on the origin is just the Euclidean circle centred on the origin with Euclidean . Prove that the hyperbolic area is .
State the Gauss-Bonnet theorem for the area of a hyperbolic triangle. Given a hyperbolic triangle and an interior point , show that the distance from to the nearest side is at most .
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3.II.12A
2005 commentDescribe geometrically the stereographic projection map from the unit sphere to the extended complex plane , positioned equatorially, and find a formula for .
Show that any Möbius transformation on has one or two fixed points. Show that the Möbius transformation corresponding (under the stereographic projection map) to a rotation of through a non-zero angle has exactly two fixed points and , where . If now is a Möbius transformation with two fixed points and satisfying , prove that either corresponds to a rotation of , or one of the fixed points, say , is an attractive fixed point, i.e. for as .
[You may assume the fact that any rotation of corresponds to some Möbius transformation of under the stereographic projection map.]
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4.II.12A
2005 commentGiven a parametrized smooth embedded surface , where is an open subset of with coordinates , and a point , define what is meant by the tangent space at , the unit normal at , and the first fundamental form
[You need not show that your definitions are independent of the parametrization.]
The second fundamental form is defined to be
where and . Prove that the partial derivatives of (considered as a vector-valued function of ) are of the form , , where
Explain briefly the significance of the determinant .
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1.II.11B
2005 commentLet be a sequence of continuous real-valued functions defined on a set . Suppose that the functions converge uniformly to a function . Prove that is continuous on .
Show that the series defines a continuous function on the half-open interval .
[Hint: You may assume the convergence of standard series.]
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2.I.3B
2005 commentDefine uniform continuity for a real-valued function defined on an interval in .
Is a uniformly continuous function on the interval necessarily bounded?
Is uniformly continuous on ?
Is uniformly continuous on ?
Justify your answers.
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2.II.13B
2005 commentUse the standard metric on in this question.
(i) Let be a nonempty closed subset of and a point in . Show that there is a point which minimizes the distance to , in the sense that for all .
(ii) Suppose that the set in part (i) is convex, meaning that contains the line segment between any two of its points. Show that point described in part (i) is unique.
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3.I.3B
2005 commentLet be a function. What does it mean to say that is differentiable at a point in ? Show that if is differentiable at , then is continuous at .
For each of the following functions, determine whether or not it is differentiable at . Justify your answers.
(i)
(ii)
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3.II.13B
2005 commentLet be a real-valued differentiable function on an open subset of . Assume that and that for all and is also in . Suppose that is homogeneous of degree , in the sense that for all and . By means of the Chain Rule or otherwise, show that
for all . (Here denotes the derivative of at , viewed as a linear map .)
Conversely, show that any differentiable function on with for all must be homogeneous of degree .
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4.I 3 B
2005 commentLet be the vector space of continuous real-valued functions on . Show that the function
defines a norm on .
For , let . Is a convergent sequence in the space with this norm? Justify your answer.
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4.II.13B
2005 commentLet be a continuous function. Let be the maximum value of . Suppose there is a constant such that
for all and . Let . Show that there is a unique function such that
and
[Hint: First show that the differential equation with its initial condition is equivalent to the integral equation
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1.II.12A
2005 commentSuppose that and are metric spaces. Show that the definition
defines a metric on the product , under which the projection map is continuous.
If is compact, show that every sequence in has a subsequence converging to a point of . Deduce that the projection map then has the property that, for any closed subset , the image is closed in . Give an example to show that this fails if is not assumed compact.
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2.I.4A
2005 commentLet be a topological space. Suppose that are connected subsets of with non-empty for all . Prove that
is connected. If each is path-connected, prove that is path-connected.
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3.I.4A
2005 commentShow that a topology is determined on the real line by specifying that a nonempty subset is open if and only if it is a union of half-open intervals , where are real numbers. Determine whether is Hausdorff.
Let denote the cofinite topology on (that is, a non-empty subset is open if and only if its complement is finite). Prove that the identity map induces a continuous .
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4.II.14A
2005 commentLet be a metric space, and a non-empty closed subset of . For , set
Prove that is a continuous function of , and that it is strictly positive for .
A topological space is called normal if for any pair of disjoint closed subsets , there exist disjoint open subsets . By considering the function
or otherwise, deduce that any metric space is normal.
Suppose now that is a normal topological space, and that are disjoint closed subsets in . Prove that there exist open subsets , whose closures are disjoint. In the case when with the standard metric topology, and , find explicit open subsets with the above property.
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1.I.3F
2005 commentState the Cauchy integral formula.
Using the Cauchy integral formula, evaluate
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1.II.13F
2005 commentDetermine a conformal mapping from to the complex unit disc
[Hint: A standard method is first to map to , then to the complex right half-plane and, finally, to
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2.II.14F
2005 commentLet be a rational function, where and has no real zeros. Using the calculus of residues, write a general expression for
in terms of residues and briefly sketch its proof.
Evaluate explicitly the integral
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3.II.14A
2005 commentState the Cauchy integral formula, and use it to deduce Liouville's theorem.
Let be a meromorphic function on the complex plane such that is bounded outside some disc (for some fixed integer ). By considering Laurent expansions, or otherwise, show that is a rational function in .
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4.I.4A
2005 commentLet be a closed path, where all paths are assumed to be piecewise continuously differentiable, and let be a complex number not in the image of . Write down an expression for the winding number in terms of a contour integral. From this characterization of the winding number, prove the following properties:
(a) If and are closed paths not passing through zero, and if is defined by for all , then
(b) If is a closed path whose image is contained in , then .
(c) If and are closed paths and is a complex number, not in the image of either path, such that
for all , then .
[You may wish here to consider the path defined by .]
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3.I.5F
2005 commentDefine a harmonic function and state when the harmonic functions and are conjugate
Let and be two pairs of harmonic conjugate functions. Prove that are also harmonic conjugate.
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4.II.15F
2005 commentDetermine the Fourier expansion of the function , where , in the two cases where is an integer and is a real non-integer.
Using the Parseval identity in the case , find an explicit expression for the sum
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1.II.14E
2005 commentFind the Fourier Series of the function
Find the solution of the Poisson equation in two dimensions inside the unit disk
subject to the boundary condition .
[Hint: The general solution of is ]
From the solution, show that
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2.I.5E
2005 commentConsider the differential equation for in
subject to boundary conditions , and . Find the Green function such that the solution for is given by
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2.II.15E
2005 commentWrite down the Euler-Lagrange equation for the variational problem for
with boundary conditions , where is a given positive constant. Show that if does not depend explicitly on , i.e. , then the equation has a first integral
where is a constant.
An axisymmetric soap film is formed between two circular rings at . Find the equation governing the shape which minimizes the surface area. Show that the shape takes the form
Show that there exist no solution if , where is the unique positive solution of .
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3.I.6E
2005 commentDescribe briefly the method of Lagrangian multipliers for finding the stationary points of a function subject to a constraint .
Use the method to find the stationary values of subject to the constraint
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3.II.15H
2005 commentObtain the power series solution about of
and show that regular solutions , which are polynomials of degree , are obtained only if Show that the polynomial must be even or odd according to the value of .
Show that
for some .
Using the identity
and considering an expansion show that
if we assume .
By considering
determine the coefficient .
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4.I.5H
2005 commentShow how the general solution of the wave equation for ,
can be expressed as
Show that the boundary conditions relate the functions and and require them to be periodic with period .
Show that, with these boundary conditions,
and that this is a constant independent of .
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4.II.16H
2005 commentDefine an isotropic tensor and show that are isotropic tensors.
For a unit vector and the area element on the unit sphere show that
is an isotropic tensor for any . Hence show that
for some which should be determined.
Explain why
where is the region inside the unit sphere.
[The general isotropic tensor of rank 4 has the form ]
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1.II.15G
2005 commentThe wave function of a particle of mass that moves in a one-dimensional potential well satisfies the Schrödinger equation with a potential that is zero in the region and infinite elsewhere,
Determine the complete set of normalised energy eigenfunctions for the particle and show that the energy eigenvalues are
where is a positive integer.
At time the wave function is
in the region , and zero otherwise. Determine the possible results for a measurement of the energy of the system and the relative probabilities of obtaining these energies.
In an experiment the system is measured to be in its lowest possible energy eigenstate. The width of the well is then doubled while the wave function is unaltered. Calculate the probability that a later measurement will find the particle to be in the lowest energy state of the new potential well.
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2.II.16G
2005 commentA particle of mass moving in a one-dimensional harmonic oscillator potential satisfies the Schrödinger equation
where the Hamiltonian is given by
The operators and are defined by
where and is the usual momentum operator. Show that .
Express and in terms of and and, hence or otherwise, show that can be written in the form
Show, for an arbitrary wave function , that and hence that the energy of any state satisfies the bound
Hence, or otherwise, show that the ground state wave function satisfies and that its energy is given by
By considering acting on , and so on, show that states of the form
are also eigenstates and that their energies are given by
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3.I.7G
2005 commentThe wave function is a solution of the time-dependent Schrödinger equation for a particle of mass in a potential ,
where is the Hamiltonian. Define the expectation value, , of any operator .
At time can be written as a sum of the form
where is a complete set of normalized eigenfunctions of the Hamiltonian with energy eigenvalues and are complex coefficients that satisfy . Find for . What is the probability of finding the system in a state with energy at time ?
Show that the expectation value of the energy is independent of time.
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3.II.16G
2005 commentA particle of mass moves in two dimensions in an axisymmetric potential. Show that the time-independent Schrödinger equation can be separated in polar coordinates. Show that the angular part of the wave function has the form , where is the angular coordinate and is an integer. Suppose that the potential is zero for , where is the radial coordinate, and infinite otherwise. Show that the radial part of the wave function satisfies
where . What conditions must satisfy at and ?
Show that, when , the equation has the solution , where if is odd and
if is even
Deduce the coefficients and in terms of . By truncating the series expansion at order , estimate the smallest value of at which the is zero. Hence give an estimate of the ground state energy.
[You may use the fact that the Laplace operator is given in polar coordinates by the expression
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4.I.6G
2005 commentDefine the commutator of two operators, and . In three dimensions angular momentum is defined by a vector operator with components
Show that and use this, together with permutations, to show that , where denotes any of the directions .
At a given time the wave function of a particle is given by
Show that this is an eigenstate of with eigenvalue equal to .
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1.II.16H
2005 commentFor a static charge density show that the energy may be expressed as
where is the electrostatic potential and is the electric field.
Determine the scalar potential and electric field for a sphere of radius with a constant charge density . Also determine the total electrostatic energy.
In a nucleus with protons the volume is proportional to . Show that we may expect the electric contribution to energy to be proportional to .
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2.I.6H
2005 commentWrite down Maxwell's equations in the presence of a charge density and current density . Show that it is necessary that satisfy a conservation equation.
If are zero outside a fixed region show that the total charge inside is a constant and also that
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2.II.17H
2005 commentAssume the magnetic field
where is a unit vector in the vertical direction. Show that this satisfies the expected equations for a static magnetic field in vacuum.
A circular wire loop, of radius , mass and resistance , lies in a horizontal plane with its centre on the -axis at a height and there is a magnetic field given by . Calculate the magnetic flux arising from this magnetic field through the loop and also the force acting on the loop when a current is flowing around the loop in a clockwise direction about the -axis.
Obtain an equation of motion for the height when the wire loop is falling under gravity. Show that there is a solution in which the loop falls with constant speed which should be determined. Verify that in this situation the rate at which heat is generated by the current flowing in the loop is equal to the rate of loss of gravitational potential energy. What happens when ?
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3.II.17H
2005 commentIf are solutions of Maxwell's equations in a region without any charges or currents show that are also solutions.
At the boundary of a perfect conductor with normal briefly explain why
Electromagnetic waves inside a perfectly conducting tube with axis along the -axis are given by the real parts of complex solutions of Maxwell's equations of the form
Suppose . Show that we can find a solution in this case in terms of a function where
so long as satisfies
for suitable . Show that the boundary conditions are satisfied if on the surface of the tube.
Obtain a similar solution with but show that the boundary conditions are now satisfied if the normal derivative on the surface of the tube.
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4.I.7H
2005 commentFor a static current density show that we may choose the vector potential so that
For a loop , centred at the origin, carrying a current show that
[You may assume
and for fixed vectors
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1.I.4G
2005 commentThe four-velocity of a particle of rest mass is defined by , where is the proper time (the time as measured in the particle's rest frame). Derive the expression for each of the four components of in terms of the components of the three-velocity and the speed of light, .
Show that for an appropriately defined scalar product.
The four-momentum, , of a particle of rest mass defines a relativistic generalisation of energy and momentum. Show that the standard non-relativistic expressions for the momentum and kinetic energy of a particle with mass travelling with velocity are obtained in the limit . Show also how the concept of a rest energy equal to emerges.
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2.I.7G
2005 commentBob and Alice are twins. Bob accelerates rapidly away from Earth in a rocket that travels in a straight line until it reaches a velocity relative to the Earth. It then travels with constant for a long time before reversing its engines and decelerating rapidly until it is travelling at a velocity relative to the Earth. After a further long period of time the rocket returns to Earth, decelerating rapidly until it is at rest. Alice remains on Earth throughout. Sketch the space-time diagram that describes Bob's world-line in Alice's rest frame, assuming that the periods of acceleration and deceleration are negligibly small compared to the total time, explain carefully why Bob ages less than Alice between his departure and his return and show that
where is the time by which Bob has aged and is the time by which Alice has aged.
Indicate on your diagram how Bob sees Alice aging during his voyage.
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4.II.17G
2005 commentObtain the Lorentz transformations that relate the coordinates of an event measured in one inertial frame to those in another inertial frame moving with velocity along the axis. Take care to state the assumptions that lead to your result.
A star is at rest in a three-dimensional coordinate frame that is moving at constant velocity along the axis of a second coordinate frame . The star emits light of frequency , which may considered to be a beam of photons. A light ray from the star to the origin in is a straight line that makes an angle with the axis. Write down the components of the four-momentum of a photon in this light ray.
The star is seen by an observer at rest at the origin of at time , when the origins of the coordinate frames and coincide. The light that reaches the observer moves along a line through the origin that makes an angle to the axis and has frequency . Make use of the Lorentz transformations between the four-momenta of a photon in these two frames to determine the relation
where is the observed wavelength of the photon and is the wavelength in the star's rest frame.
Comment on the form of this result in the special cases with and .
[You may assume that the energy of a photon of frequency momentum is a three-vector of magnitude
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1.I.5E
2005 commentExplain how a streamfunction can be used to represent in Cartesian Coordinates an incompressible flow in two dimensions. Show that the streamlines are given by const.
Consider the two-dimensional incompressible flow
(a) Find the streamfunction, and hence the streamlines at .
(b) Find the path of a fluid particle released at from . For what value of does the particle not tend to infinity?
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1.II.17E
2005 commentState Bernoulli's expression for the pressure in an unsteady potential flow with conservative force .
A spherical bubble in an incompressible liquid of density has radius . If the pressure far from the bubble is and inside the bubble is , show that
Calculate the kinetic energy in the flow outside the bubble, and hence show that
where is the volume of the bubble.
If , show that
where when .
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2.I.8E
2005 commentFor a steady flow of an incompressible fluid of density , show that
where is the vorticity and is to be found. Deduce that is constant along streamlines.
Now consider a flow in the -plane described by a streamfunction . Evaluate and deduce from that
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3.II.18E
2005 commentConsider the velocity potential in plane polar coordinates
Find the velocity field and show that it corresponds to flow past a cylinder with circulation and uniform flow at large distances.
Find the distribution of pressure over the surface of the cylinder. Hence find the and components of the force on the cylinder
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4.II.18E
2005 commentA fluid of density occupies the region and a second fluid of density occupies the region . State the equations and boundary conditions that are satisfied by the corresponding velocity potentials and and pressures and when the system is perturbed so that the interface is at and the motion is irrotational.
Seek a set of linearised equations and boundary conditions when the disturbances are proportional to , and derive the dispersion relation
where is the gravitational acceleration.
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1.I.6F
2005 commentDetermine the Cholesky factorization (without pivoting) of the matrix
where is a real parameter. Hence, find the range of values of for which the matrix is positive definite.
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2.II.18F
2005 comment(a) Let be a set of polynomials orthogonal with respect to some inner product in the interval . Write explicitly the least-squares approximation to by an th-degree polynomial in terms of the polynomials .
(b) Let an inner product be defined by the formula
Determine the th degree polynomial approximation of with respect to this inner product as a linear combination of the underlying orthogonal polynomials.
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3.II.19F
2005 commentGiven real , we consider the matrix
Construct the Jacobi and Gauss-Seidel iteration matrices originating in the solution of the linear system .
Determine the range of real for which each iterative procedure converges.
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4.I.8F
2005 commentDefine Gaussian quadrature.
Evaluate the coefficients of the Gaussian quadrature of the integral
which uses two function evaluations.
-
2005
commentThe fast-food chain McGonagles have three sizes of their takeaway haggis, Large, Jumbo and Soopersize. A survey of 300 randomly selected customers at one branch choose 92 Large, 89 Jumbo and 119 Soopersize haggises.
Is there sufficient evidence to reject the hypothesis that all three sizes are equally popular? Explain your answer carefully.
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1.II.18D
2005 commentIn the context of hypothesis testing define the following terms: (i) simple hypothesis; (ii) critical region; (iii) size; (iv) power; and (v) type II error probability.
State, without proof, the Neyman-Pearson lemma.
Let be a single observation from a probability density function . It is desired to test the hypothesis
with and , where is the distribution function of the standard normal, .
Determine the best test of size , where , and express its power in terms of and .
Find the size of the test that minimizes the sum of the error probabilities. Explain your reasoning carefully.
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2.II.19D
2005 commentLet be a random sample from a probability density function , where is an unknown real-valued parameter which is assumed to have a prior density . Determine the optimal Bayes point estimate of , in terms of the posterior distribution of given , when the loss function is
where and are given positive constants.
Calculate the estimate explicitly in the case when is the density of the uniform distribution on and .
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3.I.8D
2005 commentLet be a random sample from a normal distribution with mean and variance , where and are unknown. Derive the form of the size- generalized likelihood-ratio test of the hypothesis against , and show that it is equivalent to the standard -test of size .
[You should state, but need not derive, the distribution of the test statistic.]
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4.II.19D
2005 commentLet be observations satisfying
where are independent random variables each with the distribution. Here are known but and are unknown.
(i) Determine the maximum-likelihood estimates of .
(ii) Find the distribution of .
(iii) By showing that and are independent, or otherwise, determine the joint distribution of and .
(iv) Explain carefully how you would test the hypothesis against .
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1.I.8D
2005 commentConsider the problem:
where satisfy .
Formulate the dual of this problem and state necessary and sufficient conditions for optimality.
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2.I.9D
2005 commentExplain what is meant by a two-person zero-sum game with payoff matrix .
Show that the problems of the two players may be expressed as a dual pair of linear programming problems. State without proof a set of sufficient conditions for a pair of strategies for the two players to be optimal.
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3.II.20D
2005 commentConsider the linear programming problem
(a) After adding slack variables and and performing one pivot in the simplex algorithm the following tableau is obtained:
\begin{tabular}{c|rrrrrr|r} & & & & & & & \ \hline & 0 & 1 & & 1 & 0 & 0 & 11 \ & 0 & & & 0 & 1 & & \ & 1 & & & 0 & 0 & & \ \hline Payoff & 0 & & & 0 & 0 & & \end{tabular}
Complete the solution of the problem using the simplex algorithm.
(b) Obtain the dual problem and identify its optimal solution from the optimal tableau in (a).
(c) Suppose that the right-hand sides in the constraints to the original problem are changed from to . Give necessary and sufficient conditions on which ensure that the optimal solution to the dual obtained in (b) remains optimal for the dual for the amended problem.
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4.II.20D
2005 commentDescribe the Ford-Fulkerson algorithm for finding a maximal flow from a source to a sink in a directed network with capacity constraints on the arcs. Explain why the algorithm terminates at an optimal flow when the initial flow and the capacity constraints are rational.
Illustrate the algorithm by applying it to the problem of finding a maximal flow from to in the network below.

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1.II.19D
2005 commentEvery night Lancelot and Guinevere sit down with four guests for a meal at a circular dining table. The six diners are equally spaced around the table and just before each meal two individuals are chosen at random and they exchange places from the previous night while the other four diners stay in the same places they occupied at the last meal; the choices on successive nights are made independently. On the first night Lancelot and Guinevere are seated next to each other.
Find the probability that they are seated diametrically opposite each other on the th night at the round table, .
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2.II.20D
2005 commentConsider a Markov chain with state space and transition probabilities given by
with , otherwise, where and .
For each , let
that is, the probability that the chain ever hits the state 0 given that it starts in state . Write down the equations satisfied by the probabilities and hence, or otherwise, show that they satisfy a second-order recurrence relation with constant coefficients. Calculate for each .
Determine for each value of , whether the chain is transient, null recurrent or positive recurrent and in the last case calculate the stationary distribution.
[Hint: When the chain is positive recurrent, the stationary distribution is geometric.]
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3.I.9D
2005 commentProve that if two states of a Markov chain communicate then they have the same period.
Consider a Markov chain with state space and transition probabilities determined by the matrix
Identify the communicating classes of the chain and for each class state whether it is open or closed and determine its period.
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4.I.9D
2005 commentProve that the simple symmetric random walk in three dimensions is transient.
[You may wish to recall Stirling's formula: ]
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1.I.1H
2005 commentDefine the Legendre symbol . Prove that, if is an odd prime, then
Use the law of quadratic reciprocity to calculate .
[You may use the Gauss Lemma without proof.]
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2.I.1H
2005 commentRecall that, if is an odd prime, a primitive root modulo is a generator of the cyclic (multiplicative) group . Let be an odd prime of the form ; show that is a primitive root if and only if is not a quadratic residue mod . Use this result to prove that 7 is a primitive root modulo every such prime.
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3.I.1H
2005 commentLet be the number of primes . State the Legendre formula, and prove that
[You may use the formula
without proof.]
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3.II.11H
2005 commentShow that there are exactly two reduced positive definite integer binary quadratic forms with discriminant ; write these forms down.
State a criterion for an odd integer to be properly represented by a positive definite integer binary quadratic form of given discriminant .
Describe, in terms of congruences modulo 20, which primes other than 2,5 are properly represented by the form , and justify your answer.
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4.I.1H
2005 commentIf is an odd integer and is an integer prime with , state what it means for to be a pseudoprime to the base . What is a Carmichael number? State a criterion for to be a Carmichael number and use the criterion to show that:
(i) Every Carmichael number is the product of at least three distinct primes.
(ii) 561 is a Carmichael number.
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4.II.11H
2005 comment(a) Let be a non-square integer. Describe the integer solutions of the Pell equation in terms of the convergents to . Show that the set of integer solutions forms an abelian group. Denote the addition law in this group by ; given solutions and , write down an explicit formula for . If is a solution, write down an explicit formula for in the group law.
(b) Find the continued fraction expansion of . Find the smallest solution in integers of the Pell equation . Use the formula in Part (a) to compute .
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1.I.2F
2005 commentProve that is irrational.
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1.II.11F
2005 commentState and prove a discrete form of Brouwer's theorem, concerning colourings of points in triangular grids. Use it to deduce that there is no continuous retraction from a disc to its boundary.
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2005
comment(i) Let be an algebraic number and let and be integers with . What does Liouville's theorem say about and ?
(ii) Let and be integers with . Prove that
[In (ii), you may not use Liouville's theorem unless you prove it.]
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2.II.11F
2005 comment(i) State the Baire category theorem. Deduce from it a statement about nowhere dense sets.
(ii) Let be the set of all real numbers with decimal expansions consisting of the digits 4 and 5 only. Prove that there is a real number that cannot be written in the form with and rational.
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2005
commentLet and let be real numbers such that
for every polynomial of degree less than . Prove the following three facts.
(i) for every .
(ii) .
(iii) The numbers are the roots of the Legendre polynomial of degree .
[You may assume standard orthogonality properties of the Legendre polynomials.]
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4.I.2F
2005 comment(i) Let be a domain, let be an analytic function and let . What does Taylor's theorem say about and ?
(ii) Let be the square consisting of all complex numbers such that
and let be a complex number not belonging to . Prove that the function can be uniformly approximated on by polynomials.
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1.I.3G
2005 commentLet be a subgroup of the group of isometries of the Euclidean plane. What does it mean to say that is discrete?
Supposing that is discrete, show that the subgroup of consisting of all translations in is generated by translations in at most two linearly independent vectors in . Show that there is a homomorphism with kernel .
Draw, and briefly explain, pictures which illustrate two different possibilities for when is isomorphic to the additive group .
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1.II.12G
2005 commentWhat is the limit set of a subgroup of Möbius transformations?
Suppose that is complicated and has no finite orbit in . Prove that the limit set of is infinite. Can the limit set be countable?
State Jørgensen's inequality, and deduce that not every two-generator subgroup of Möbius transformations is discrete. Briefly describe two examples of discrete two-generator subgroups, one for which the limit set is connected and one for which it is disconnected.
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2.I.3G
2005 commentDescribe the geodesics in the disc model of the hyperbolic plane .
Define the area of a region in . Compute the area of a hyperbolic circle of radius from the definition just given. Compute the circumference of a hyperbolic circle of radius , and check explicitly that .
How could you define geometrically if you lived in ? Briefly justify your answer.
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3.I.3G
2005 commentBy considering fixed points in , prove that any complex Möbius transformation is conjugate either to a map of the form for some or to . Deduce that two Möbius transformations (neither the identity) are conjugate if and only if .
Does every Möbius transformation also have a fixed point in ? Briefly justify your answer.
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4.I.3G
2005 commentShow that a set with Hausdorff dimension strictly less than one is totally disconnected.
What does it mean for a Möbius transformation to pair two discs? By considering a pair of disjoint discs and a pair of tangent discs, or otherwise, explain in words why there is a 2-generator Schottky group with limit set which has Hausdorff dimension at least 1 but which is not homeomorphic to a circle.
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4.II.12G
2005 commentFor real and , give a careful definition of the -dimensional Hausdorff measure of and of the Hausdorff dimension of .
For , suppose is a similarity with contraction factor . Prove there is a unique non-empty compact invariant set for the . State a formula for the Hausdorff dimension of , under an assumption on the you should state.
Hence show the Hausdorff dimension of the fractal given by iterating the scheme below (at each stage replacing each edge by a new copy of the generating template) is .

[Numbers denote lengths]
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1.I.4J
2005 commentBriefly describe the methods of Shannon-Fano and Huffman for economical coding. Illustrate both methods by finding decipherable binary codings in the case where messages are emitted with probabilities . Compute the expected word length in each case.
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2.I.4J
2005 commentWhat is a linear binary code? What is the weight of a linear binary code Define the bar product of two binary linear codes and , stating the conditions that and must satisfy. Under these conditions show that
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2.II.12J
2005 commentWhat does it means to say that is a linear feedback shift register? Let be a stream produced by such a register. Show that there exist with such that for all .
Explain and justify the Berlekamp-Massey method for 'breaking' a cipher stream arising from a linear feedback register of unknown length.
Let be three streams produced by linear feedback registers. Set
Show that is also a stream produced by a linear feedback register. Sketch proofs of any theorems that you use.
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3.I.4J
2005 commentBriefly explain how and why a signature scheme is used. Describe the el Gamal scheme.
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3.II.12J
2005 commentDefine a cyclic code. Define the generator and check polynomials of a cyclic code and show that they exist.
Show that Hamming's original code is a cyclic code with check polynomial . What is its generator polynomial? Does Hamming's original code contain a subcode equivalent to its dual?
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4.I.4J
2005 commentWhat does it mean to transmit reliably at rate through a binary symmetric channel (BSC) with error probability ? Assuming Shannon's second coding theorem, compute the supremum of all possible reliable transmission rates of a BSC. What happens if (i) is very small, (ii) , or (iii) ?
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1.I.5I
2005 commentSuppose that are independent random variables, and that has probability density function
Assume that and that there is a known link function such that
where are known -dimensional vectors and is an unknown -dimensional parameter. Show that and that, if is the log-likelihood function from the observations , then
where is to be defined.
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1.II.13I
2005 commentThe Independent, June 1999 , under the headline 'Tourists get hidden costs warnings' gave the following table of prices in pounds, called 'How the resorts compared'.
Here the column headings are, respectively: Three-course meal, Bottle of Beer, Suntan Lotion, Taxi (5km), Film (24 exp), Car Hire (per week). Interpret the commands, and explain how to interpret the corresponding (slightly abbreviated) output given below. Your solution should include a careful statement of the underlying statistical model, but you may quote without proof any distributional results required.

Residual standard error: on 65 degrees of freedom
Multiple R-Squared:
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2.I.5I
2005 commentYou see below three commands, and the corresponding output (which is slightly abbreviated). Explain the effects of the commands. How is the deviance defined, and why do we have d.f. in this case? Interpret the numerical values found in the output.

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3.I.5I
2005 commentConsider the model , where is an -dimensional observation vector, is an matrix of rank is an -dimensional vector with components , and are independently and normally distributed, each with mean 0 and variance
(a) Let be the least-squares estimator of . Show that
and find the distribution of .
(b) Define . Show that has distribution , where is a matrix that you should define.
[You may quote without proof any results you require about the multivariate normal distribution.]
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4.I.5I
2005 commentYou see below five commands, and the corresponding output (which is slightly abbreviated). Without giving any mathematical proofs, explain the purpose of these commands, and interpret the output.

Residual deviance: on 2 degrees of freedom
Number of Fisher Scoring iterations: 4
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4.II.13I
2005 comment(i) Suppose that are independent random variables, and that has probability density function
where
and are given -dimensional vectors, and is known.
Show that and that .
(ii) Find the equation for , the maximum likelihood estimator of , and suggest an iterative scheme for its solution.
(iii) If , and , find the large-sample distribution of . Write your answer in terms of and , where are defined by
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1.I.6E
2005 commentConsider a biological system in which concentrations and satisfy
where and are positive and monotonically decreasing functions of their arguments, so that represses the synthesis of and vice versa.
(a) Suppose the functions and are bounded. Sketch the phase plane and explain why there is always at least one steady state. Show that if there is a steady state with
then the system is multistable.
(b) If and , where and are positive constants, what values of and allow the system to display multistability for some value of ?
Can and generate multistability? Explain your answer carefully.
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2.I.6E
2005 commentConsider a system with stochastic reaction events
where and are rate constants.
(a) State or derive the exact differential equation satisfied by the average number of molecules . Assuming that fluctuations are negligible, approximate the differential equation to obtain the steady-state value of .
(b) Using this approximation, calculate the elasticity , the average lifetime , and the average chemical event size (averaged over fluxes).
(c) State the stationary Fluctuation Dissipation Theorem for the normalised variance . Hence show that
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2.II.13E
2005 commentConsider the reaction-diffusion system
for an activator and inhibitor , where and are degradation rate constants and and are diffusion rate constants.
(a) Find a suitably scaled time and length such that
and find expressions for and .
(b) Show that the Jacobian matrix for small spatially homogenous deviations from a nonzero steady state of is
and find the values of for which the steady state is stable.
[Hint: The eigenvalues of a real matrix both have positive real parts iff the matrix has a positive trace and determinant.]
(c) Derive linearised ordinary differential equations for the amplitudes and of small spatially inhomogeneous deviations from a steady state of that are proportional to , where is a constant.
(d) Assuming that the system is stable to homogeneous perturbations, derive the condition for inhomogeneous instability. Interpret this condition in terms of how far activator and inhibitor molecules diffuse on average before they are degraded.
(e) Calculate the lengthscale of disturbances that are expected to be observed when the condition for inhomogeneous instability is just satisfied. What are the dominant mechanisms for stabilising disturbances on lengthscales (i) much less than and (ii) much greater than ?
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2005
commentLet be the concentration of a binary master sequence of length and let be the total concentration of all mutant sequences. Master sequences try to self-replicate at a total rate , but each independent digit is only copied correctly with probability . Mutant sequences self-replicate at a total rate , where , and the probability of mutation back to the master sequence is negligible.
(a) The evolution of is given by
Write down the corresponding equation for and derive a differential equation for the master-to-mutant ratio .
(b) What is the maximum length for which there is a positive steady-state value of ? Is the positive steady state stable when it exists?
(c) Obtain a first-order approximation to assuming that both and , where the selection coefficient is defined by .
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3.II.13E
2005 commentProtein synthesis by RNA can be represented by the stochastic system
in which is an environmental variable corresponding to the number of RNA molecules per cell and is a system variable, with birth rate proportional to , corresponding to the number of protein molecules.
(a) Use the normalized stationary Fluctuation-Dissipation Theorem (FDT) to calculate the (exact) normalized stationary variances and in terms of the averages and .
(b) Separate into an intrinsic and an extrinsic term by considering the limits when does not fluctuate (intrinsic), and when responds deterministically to changes in (extrinsic). Explain how the extrinsic term represents the magnitude of environmental fluctuations and time-averaging.
(c) Assume now that the birth rate of is changed from the "constitutive" mechanism in (1) to a "negative feedback" mechanism , where is a monotonically decreasing function of . Use the stationary FDT to approximate in terms of . Apply your answer to the case .
[Hint: To reduce the algebra introduce the elasticity , where and are the death and birth rates of respectively.]
(d) Explain the extrinsic term for the negative feedback system in terms of environmental fluctuations, time-averaging, and static susceptibility.
(e) Explain why the FDT is exact for the constitutive system but approximate for the feedback system. When, generally speaking, does the FDT approximation work well?
(f) Consider the following three experimental observations: (i) Large changes in have no effect on ; (ii) When is perturbed by from its stationary average, perturbations are corrected more rapidly in the feedback system than in the constitutive system; (iii) The feedback system displays lower values than the constitutive system.
What does (i) imply about the relative importance of the noise terms? Can (ii) be directly explained by (iii), i.e., does rapid adjustment reduce noise? Justify your answers.
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4.I.6E
2005 commentThe output of a linear perceptron is given by , where is a vector of weights connecting a fluctuating input vector to an output unit. The weights are given random initial values and are then updated according to a learning rule that has a time-constant much greater than the fluctuation timescale of the inputs.
(a) Find the behaviour of for each of the following two rules (i) (ii) , where is a positive constant.
(b) Consider a third learning rule
Show that in a steady state the vector of weights satisfies the eigenvalue equation
where the matrix and eigenvalue should be identified.
(c) Comment briefly on the biological implications of the three rules.
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1.I.7B
2005 commentState Dulac's Criterion and the Poincaré-Bendixson Theorem regarding the existence of periodic solutions to the dynamical system in . Hence show that
has no periodic solutions if and at least one periodic solution if .
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1.II.14B
2005 commentConsider the equations
as a function of the parameter . Find the fixed points and plot their location in the plane. Hence, or otherwise, deduce that there are bifurcations at and .
Investigate the bifurcation at by making the substitutions and . Find the equation of the extended centre manifold to second order. Find the evolution equation on the centre manifold to second order, and determine the stability of its fixed points.
Show which branches of fixed points in the plane are stable and which are unstable, and state, without calculation, the type of bifurcation at . Hence sketch the structure of the phase plane very near the origin for in the cases (i) and (ii) .
The system is perturbed to , where , with still. Sketch the possible changes to the bifurcation diagram near and . [Calculation is not required.]
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2.I.7B
2005 commentDefine Lyapunov stability and quasi-asymptotic stability of a fixed point of a dynamical system .
By considering a Lyapunov function of the form , show that the origin is an asymptotically stable fixed point of
[Lyapunov's Second Theorem may be used without proof, provided you show that its conditions apply.]
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2.II.14B
2005 commentProve that if a continuous map of an interval into itself has a periodic orbit of period three then it also has periodic orbits of least period for all positive integers .
Explain briefly why there must be at least two periodic orbits of least period
[You may assume without proof:
(i) If and are non-empty closed bounded intervals such that then there is a closed bounded interval such that .
(ii) The Intermediate Value Theorem.]
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3.I.7B
2005 commentDefine the stable and unstable invariant subspaces of the linearisation of a dynamical system at a saddle point located at the origin in . How, according to the Stable Manifold Theorem, are the stable and unstable manifolds related to the invariant subspaces?
Calculate the stable and unstable manifolds, correct to cubic order, for the system
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4.I.7B
2005 commentFind and classify the fixed points of the system
Sketch the phase plane.
What is the -limit for the point ? Which points have as their -limit?
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1.I.8A
2005 commentExplain what is meant by the Papperitz symbol
The hypergeometric function is defined as the solution of the equation determined by the Papperitz symbol
that is analytic at and satisfies .
Show, explaining each step, that
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2.I.8A
2005 commentThe Hankel representation of the gamma function is
where the path of integration is the Hankel contour.
Use this representation to find the residue of at , where is a nonnegative integer.
Is there a pole at , where is a positive integer? Justify your answer carefully, working only from the above representation of .
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3.I.8A
2005 commentThe functions and have Laplace transforms and , and satisfy for . The convolution of and is defined by
and has Laplace transform . Prove (the convolution theorem) that .
Given that , deduce the Laplace transform of the function , where
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3.II.14A
2005 commentShow that the equation
where is constant, has solutions of the form
provided that the path is chosen so that .
(i) In the case Re , show that there is a choice of for which .
(ii) In the case , where is any integer, show that can be a finite contour and that the corresponding solution satisfies if .
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4.I.8A
2005 commentWrite down necessary and sufficient conditions on the functions and for the point to be (i) an ordinary point and (ii) a regular singular point of the equation
Show that the point is an ordinary point if and only if
where and are analytic in a neighbourhood of the origin.
Find the most general equation of the form that has a regular singular point at but no other singular points.
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4.II.14A
2005 commentTwo representations of the zeta function are
where, in the integral representation, the path is the Hankel contour and the principal branch of , for which , is to be used. State the range of for which each representation is valid.
Evaluate the integral
where is a closed path consisting of the straight line , with , and the semicircle , with , where is a positive integer.
Making use of this result and assuming, when necessary, that the integral along the curved part of is negligible when is large, derive the functional equation
for .
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1.I.9C
2005 commentA particle of mass is constrained to move on a circle of radius , centre in a horizontal plane . A second particle of mass moves on a circle of radius , centre in a horizontal plane . The two particles are connected by a spring whose potential energy is
where is the distance between the particles. How many degrees of freedom are there? Identify suitable generalized coordinates and write down the Lagrangian of the system in terms of them.
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1.II.15C
2005 comment(i) The action for a system with generalized coordinates is given by
Derive Lagrange's equations from the principle of least action by considering all paths with fixed endpoints, .
(ii) A pendulum consists of a point mass at the end of a light rod of length . The pivot of the pendulum is attached to a mass which is free to slide without friction along a horizontal rail. Choose as generalized coordinates the position of the pivot and the angle that the pendulum makes with the vertical.
Write down the Lagrangian and derive the equations of motion.
Find the frequency of small oscillations around the stable equilibrium.
Now suppose that a force acts on the pivot causing it to travel with constant acceleration in the -direction. Find the equilibrium angle of the pendulum.
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2.I.9C
2005 commentA rigid body has principal moments of inertia and and is moving under the action of no forces with angular velocity components . Its motion is described by Euler's equations
Are the components of the angular momentum to be evaluated in the body frame or the space frame?
Now suppose that an asymmetric body is moving with constant angular velocity . Show that this motion is stable if and only if is the largest or smallest principal moment.
-
3.I.9C
2005 commentDefine the Poisson bracket between two functions and on phase space. If has no explicit time dependence, and there is a Hamiltonian , show that Hamilton's equations imply
A particle with position vector and momentum has angular momentum . Compute and .
-
3.II.15C
2005 comment(i) A point mass with position and momentum undergoes one-dimensional periodic motion. Define the action variable in terms of and . Prove that an orbit of energy has period
(ii) Such a system has Hamiltonian
where is a positive constant and during the motion. Sketch the orbits in phase space both for energies and . Show that the action variable is given in terms of the energy by
Hence show that for the period of the orbit is , where is the greatest value of the momentum during the orbit.
-
4.I.9C
2005 commentDefine a canonical transformation for a one-dimensional system with coordinates . Show that if the transformation is canonical then .
Find the values of constants and such that the following transformations are canonical: (i) . (ii) .
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1.I.10D
2005 comment(a) Around after the big bang , neutrons and protons are kept in equilibrium by weak interactions such as
Show that, in equilibrium, the neutron-to-proton ratio is given by
where corresponds to the mass difference between the neutron and the proton. Explain briefly why we can neglect the difference in the chemical potentials.
(b) The ratio of the weak interaction rate which maintains (*) to the Hubble expansion rate is given by
Explain why the neutron-to-proton ratio effectively "freezes out" once , except for some slow neutron decay. Also explain why almost all neutrons are subsequently captured in ; estimate the value of the relative mass density (with ) given a final ratio .
(c) Suppose instead that the weak interaction rate were very much weaker than that described by equation . Describe the effect on the relative helium density . Briefly discuss the wider implications of this primordial helium-to-hydrogen ratio on stellar lifetimes and life on earth.
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2.I.10D
2005 comment(a) A spherically symmetric star obeys the pressure-support equation
where is the pressure at a distance from the centre, is the density, and the mass is defined through the relation . Multiply by and integrate over the total volume of the star to derive the virial theorem
where is the average pressure and is the total gravitational potential energy.
(b) Consider a white dwarf supported by electron Fermi degeneracy pressure , where is the electron mass and is the number density. Assume a uniform density , so the total mass of the star is given by where is the star radius and is the proton mass. Show that the total energy of the white dwarf can be written in the form
where are positive constants which you should determine. [You may assume that for an ideal gas .] Use this expression to explain briefly why a white dwarf is stable.
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2.II.15D
2005 comment(a) Consider a homogeneous and isotropic universe with scale factor and filled with mass density . Show how the conservation of kinetic energy plus gravitational potential energy for a test particle on the edge of a spherical region in this universe can be used to derive the Friedmann equation
where is a constant. State clearly any assumptions you have made.
(b) Now suppose that the universe was filled throughout its history with radiation with equation of state . Using the fluid conservation equation and the definition of the relative density , show that the density of this radiation can be expressed as
where is the Hubble parameter today and is the relative density today and is assumed. Show also that and hence rewrite the Friedmann equation as
where .
(c) Now consider a closed model with (or . Rewrite ( ) using the new time variable defined by
Hence, or otherwise, solve to find the parametric solution
where Recall that
Using the solution for , find the value of the new time variable today and hence deduce that the age of the universe in this model is
-
3.I.10D
2005 comment(a) Define and discuss the concept of the cosmological horizon and the Hubble radius for a homogeneous isotropic universe. Illustrate your discussion with the specific examples of the Einstein-de Sitter universe for and a de Sitter universe with constant, .
(b) Explain the horizon problem for a decelerating universe in which with . How can inflation cure the horizon problem?
(c) Consider a Tolman (radiation-filled) universe ) beginning at and lasting until today at . Estimate the horizon size today and project this lengthscale backwards in time to show that it had a physical size of about 1 metre at .
Prior to , assume an inflationary (de Sitter) epoch with constant Hubble parameter given by its value at for the Tolman universe. How much expansion during inflation is required for the observable universe today to have begun inside one Hubble radius?
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4.I.10D
2005 commentThe linearised equation for the growth of a density fluctuation in a homogeneous and isotropic universe is
where is the non-relativistic matter density, is the comoving wavenumber and is the sound speed .
(a) Define the Jeans length and discuss its significance for perturbation growth.
(b) Consider an Einstein-de Sitter universe with filled with pressure-free matter . Show that the perturbation equation can be re-expressed as
By seeking power law solutions, find the growing and decaying modes of this equation.
(c) Qualitatively describe the evolution of non-relativistic matter perturbations in the radiation era, , when . What feature in the power spectrum is associated with the matter-radiation transition?
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4.II.15D
2005 commentFor an ideal gas of bosons, the average occupation number can be expressed as
where has been included to account for the degeneracy of the energy level . In the approximation in which a discrete set of energies is replaced with a continuous set with momentum , the density of one-particle states with momentum in the range to is . Explain briefly why
where is the volume of the gas. Using this formula with equation , obtain an expression for the total energy density of an ultra-relativistic gas of bosons at zero chemical potential as an integral over . Hence show that
where is a number you should find. Why does this formula apply to photons?
Prior to a time years, the universe was filled with a gas of photons and non-relativistic free electrons and protons. Subsequently, at around years, the protons and electrons began combining to form neutral hydrogen,
Deduce Saha's equation for this recombination process stating clearly the steps required:
where is the ionization energy of hydrogen. [Note that the equilibrium number density of a non-relativistic species is given by , while the photon number density is , where
Consider now the fractional ionization , where is the baryon number of the universe and is the baryon-to-photon ratio. Find an expression for the ratio
in terms only of and constants such as and . One might expect neutral hydrogen to form at a temperature given by , but instead in our universe it forms at the much lower temperature . Briefly explain why.
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1.II.16F
2005 commentState and prove Zorn's Lemma. [You may assume Hartogs' Lemma.] Where in your argument have you made use of the Axiom of Choice?
Show that , considered as a rational vector space, has a basis.
Prove that and are isomorphic as rational vector spaces.
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2.II.16F
2005 commentGive the inductive and the synthetic definitions of ordinal addition, and prove that they are equivalent. Give an example to show that ordinal addition is not commutative.
Which of the following assertions about ordinals and are always true, and which can be false? Give proofs or counterexamples as appropriate.
(i) .
(ii) If and are limit ordinals then .
(iii) If then or .
(iv) If then or .
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3.II.16F
2005 commentState the Axiom of Foundation and the Principle of -Induction, and show that they are equivalent (in the presence of the other axioms of ZF). [You may assume the existence of transitive closures.]
Explain briefly how the Principle of -Induction implies that every set is a member of some .
For each natural number , find the cardinality of . For which ordinals is the cardinality of equal to that of the reals?
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4.II.16F
2005 commentState and prove the Completeness Theorem for Propositional Logic. [You do not need to give definitions of the various terms involved. You may assume that the set of primitive propositions is countable. You may also assume the Deduction Theorem, provided that you state it clearly.]
Where in your argument have you used the third axiom, namely
State the Compactness Theorem, and deduce it from the Completeness Theorem.
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1.II .17F
2005 commentShow that an acyclic graph has a vertex of degree at most one. Prove that a tree (that is, a connected acyclic graph) of order has size , and deduce that every connected graph of order and size is a tree.
Let be a tree of order . Show that if is a graph with then is a subgraph of , but that this need not happen if .
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2.II.17F
2005 commentBrooks' Theorem states that if is a connected graph then unless is complete or is an odd cycle. Prove the theorem for 3-connected graphs .
Let be a graph, and let . By considering a partition of that minimizes the quantity , show that there is a partition with .
By taking , show that if a graph contains no then .
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3.II.17F
2005 commentLet and be disjoint sets of vertices each. Let be a bipartite graph formed by adding edges between and randomly and independently with probability . Let be the number of edges of between the subsets and . Let . Consider three events and , as follows.
Show that and . Hence show that and so show that, almost surely, none of or occur. Deduce that, almost surely, has a matching from to .
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4.II.17F
2005 commentWrite an essay on extremal graph theory. Your essay should include the proof of at least one extremal theorem. You should state the Erdős-Stone theorem, as well as describing its proof and showing how it can be applied.
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1.II.18G
2005 commentLet be a field extension. State what it means for an element to be algebraic over . Show that is algebraic over if and only if the field is finite dimensional as a vector space over .
State what it means for a field extension to be algebraic. Show that, if is algebraic and is algebraic, then is algebraic.
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2.II.18G
2005 commentLet be a field of characteristic 0 containing all roots of unity.
(i) Let be the splitting field of the polynomial where . Show that the Galois group of is cyclic.
(ii) Suppose that is a cyclic extension of degree over . Let be a generator of the Galois group and a primitive -th root of 1 . By considering the resolvent
of elements , show that is the splitting field of a polynomial for some .
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3.II.18G
2005 commentFind the Galois group of the polynomial
over and . Hence or otherwise determine the Galois group over .
[Standard general results from Galois theory may be assumed.]
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4.II.18G
2005 comment(i) Let be the splitting field of the polynomial
over . Show that , and hence show that the Galois group of is the dihedral group of order 8 .
(ii) Let be the splitting field of the polynomial
over . Show that . Show that the Galois group of is .
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1.II.19G
2005 commentLet the finite group act on finite sets and , and denote by the associated permutation representations on the spaces of complex functions on and . Call their characters and .
(i) Show that the inner product is the number of orbits for the diagonal action of on .
(ii) Assume that , and let be the subspace of those functions whose values sum to zero. By considering , show that is irreducible if and only if the -action on is doubly transitive: this means that for any two pairs and of points in with and , there exists some with and .
(iii) Let now acting on the set . Call the set of 2element subsets of , with the natural action of . If , show that decomposes under into three irreducible representations, one of which is the trivial representation and another of which is . What happens when ?
[Hint: Consider and .]
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2.II.19G
2005 commentLet be a finite group and the set of its irreducible characters. Also choose representatives for the conjugacy classes, and denote by their centralisers.
(i) State the orthogonality and completeness relations for the .
(ii) Using Part (i), or otherwise, show that
(iii) Let be the matrix with . Prove that
(iv) Show that is either real or purely imaginary, explaining when each situation occurs.
[Hint for (iv): Consider the effect of complex conjugation on the rows of the matrix A.]
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3.II.19G
2005 commentLet be the group with 21 elements generated by and , subject to the relations and
(i) Find the conjugacy classes of .
(ii) Find three non-isomorphic one-dimensional representations of .
(iii) For a subgroup of a finite group , write down (without proof) the formula for the character of the -representation induced from a representation of .
(iv) By applying Part (iii) to the case when is the subgroup of , find the remaining irreducible characters of .
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4.II.19G
2005 comment(i) State and prove the Weyl integration formula for .
(ii) Determine the characters of the symmetric powers of the standard 2-dimensional representation of and prove that they are irreducible.
[Any general theorems from the course may be used.]
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1.II.20G
2005 commentLet where is a prime with relative traces where runs through the three quadratic subfields of , show that the algebraic integers in have the form
where are rational integers. By further computing the relative norm where , show that 4 divides
Deduce that and are even and . Hence verify that an integral basis for is
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2.II.20G
2005 commentShow that is a unit in . Show further that 2 is the square of the principal ideal in generated by .
Assuming that the Minkowski constant for is , deduce that has class number 1 .
Assuming further that is the fundamental unit in , show that the complete solution in integers of the equation is given by
Calculate the particular solution in positive integers when
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4.II.20G
2005 commentState Dedekind's theorem on the factorisation of rational primes into prime ideals.
A rational prime is said to ramify totally in a field with degree if it is the -th power of a prime ideal in the field. Show that, in the quadratic field with a squarefree integer, a rational prime ramifies totally if and only if it divides the discriminant of the field.
Verify that the same holds in the cyclotomic field , where with an odd prime, and also in the cubic field .
The cases and for the quadratic field should be carefully distinguished. It can be assumed that has a basis and discriminant , and that has a basis and discriminant
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1.II.21H
2005 comment(i) Show that if is a covering map for the torus , then is homeomorphic to one of the following: the plane , the cylinder , or the torus .
(ii) Show that any continuous map from a sphere to the torus is homotopic to a constant map.
[General theorems from the course may be used without proof, provided that they are clearly stated.]
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2.II.21H
2005 commentState the Van Kampen Theorem. Use this theorem and the fact that to compute the fundamental groups of the torus , the punctured torus , for some point , and the connected sum of two copies of .
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3.II.20H
2005 commentLet be a space that is triangulable as a simplicial complex with no -simplices. Show that any continuous map from to is homotopic to a constant map.
[General theorems from the course may be used without proof, provided they are clearly stated.]
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4.II.21H
2005 commentLet be a simplicial complex. Suppose for subcomplexes and , and let . Show that the inclusion of in induces an isomorphism if and only if the inclusion of in induces an isomorphism .
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1.II.22F
2005 commentLet be a compact Hausdorff space, and let denote the Banach space of continuous, complex-valued functions on , with the supremum norm. Define what it means for a set to be totally bounded, uniformly bounded, and equicontinuous.
Show that is totally bounded if and only if it is both uniformly bounded and equicontinuous.
Give, with justification, an example of a Banach space and a subset such that is bounded but not totally bounded.
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2.II.22F
2005 commentLet and be Banach spaces. Define what it means for a linear operator to be compact. For a linear operator , define the spectrum, point spectrum, and resolvent set of .
Now let be a complex Hilbert space. Define what it means for a linear operator to be self-adjoint. Suppose is an orthonormal basis for . Define a linear operator by setting . Is compact? Is self-adjoint? Justify your answers. Describe, with proof, the spectrum, point spectrum, and resolvent set of .
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3.II.21F
2005 commentLet be a normed vector space. Define the dual of . Define the normed vector spaces for all . [You are not required to prove that the norms you have given are indeed norms.]
Now let be such that . Show that is isometrically isomorphic to as a normed vector space. [You may assume any standard inequalities.]
Show by a similar argument that is isomorphic to . Does your argument also show that is isomorphic to ? If not, where does it fail?
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4.II.22F
2005 commentLet and be normed vector spaces. Show that a linear map is continuous if and only if it is bounded.
Now let be Banach spaces. We say that a map is bilinear
Suppose that is bilinear and is continuous in each variable separately. Show that there exists a constant such that
for all .
[Hint: For each fixed , consider the map from to .]
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1.II.23H
2005 commentLet be a lattice in generated by 1 and , where is a fixed complex number with . The Weierstrass -function is defined as a -periodic meromorphic function such that
(1) the only poles of are at points of , and
(2) there exist positive constants and such that for all , we have
Show that is uniquely determined by the above properties and that . By considering the valency of at , show that .
Show that satisfies the differential equation
for some complex constant .
[Standard theorems about doubly-periodic meromorphic functions may be used without proof provided they are accurately stated, but any properties of the -function that you use must be deduced from first principles.]
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2.II.23H
2005 commentDefine the terms function element and complete analytic function.
Let be a function element such that , for some integer , where is a complex polynomial with no multiple roots. Let be the complete analytic function containing . Show that every function element in satisfies
Describe how the non-singular complex algebraic curve
can be made into a Riemann surface such that the first and second projections define, by restriction, holomorphic maps .
Explain briefly the relation between and the Riemann surface for the complete analytic function given earlier.
[You do not need to prove the Inverse Function Theorem, provided that you state it accurately.]
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3.II.22H
2005 commentExplain what is meant by a meromorphic differential on a compact connected Riemann surface . Show that if is a meromorphic function on then defines a meromorphic differential on . Show also that if and are two meromorphic differentials on which are not identically zero then for some meromorphic function . Show that zeros and poles of a meromorphic differential are well-defined and explain, without proof, how to obtain the genus of by counting zeros and poles of .
Let be the affine curve with equation and let be the corresponding projective curve. Show that is non-singular with two points at infinity, and that extends to a meromorphic differential on .
[You may assume without proof that that the map
is onto and extends to a biholomorphic map from onto .]
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4.II.23H
2005 commentDefine what is meant by the degree of a non-constant holomorphic map between compact connected Riemann surfaces, and state the Riemann-Hurwitz formula.
Let be an elliptic curve defined by some lattice . Show that the map
is biholomorphic, and that there are four points in fixed by .
Let be the quotient surface (the topological surface obtained by identifying and , for each and let be the corresponding projection map. Denote by the complement of the four points fixed by , and let . Describe briefly a family of charts making into a Riemann surface, so that is a holomorphic map.
Now assume that the complex structure of extends to , so that is a Riemann surface, and that the map is in fact holomorphic on all of . Calculate the genus of .
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1.II.24H
2005 commentLet be a smooth map between manifolds without boundary.
(i) Define what is meant by a critical point, critical value and regular value of .
(ii) Show that if is a regular value of and , then the set is a submanifold of with .
[You may assume the inverse function theorem.]
(iii) Let be the group of all real matrices with determinant 1. Prove that is a submanifold of the set of all real matrices. Find the tangent space to at the identity matrix.
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2.II.24H
2005 commentState the isoperimetric inequality in the plane.
Let be a surface. Let and let be a geodesic circle of centre and radius ( small). Let be the length of and be the area of the region bounded by . Prove that
where is the Gaussian curvature of at and
When and is small, compare this briefly with the isoperimetric inequality in the plane.
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3.II.23H
2005 comment(i) Define geodesic curvature and state the Gauss-Bonnet theorem.
(ii) Let be a closed regular curve parametrized by arc-length, and assume that has non-zero curvature everywhere. Let be the curve given by the normal vector to . Let be the arc-length of the curve on . Show that the geodesic curvature of is given by
where and are the curvature and torsion of .
(iii) Suppose now that is a simple curve (i.e. it has no self-intersections). Show that divides into two regions of equal area.
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4.II.24H
2005 comment(i) Define what is meant by an isothermal parametrization. Let be an isothermal parametrization. Prove that
where is the mean curvature vector and .
Define what it means for to be minimal, and deduce that is minimal if and only if .
[You may assume that the mean curvature can be written as
(ii) Write . Consider the complex valued functions
Show that is isothermal if and only if .
Suppose now that is isothermal. Prove that is minimal if and only if and are holomorphic functions.
(iii) Consider the immersion given by
Find and . Show that is an isothermal parametrization of a minimal surface.
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1.II
2005 commentLet be a probability space. For , what is meant by saying that is a -system? State the 'uniqueness of extension' theorem for measures on having given values on .
For , we call independent if
If and are independent -systems, show that and are independent.
Let be independent random variables on . Show that the -fields and are independent.
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2.II.25J
2005 commentLet be a family of random variables on the common probability space . What is meant by saying that is uniformly integrable? Explain the use of uniform integrability in the study of convergence in probability and in . [Clear definitions should be given of any terms used, but proofs may be omitted.]
Let and be uniformly integrable families of random variables on . Show that the family given by
is uniformly integrable.
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2005
commentLet be a measure space. For a measurable function , and , let . Let be the space of all such with . Explain what is meant by each of the following statements:
(a) A sequence of functions is Cauchy in .
(b) is complete.
Show that is complete for .
Take the Borel -field of , and the Lebesgue measure on . For , determine which if any of the following sequences of functions are Cauchy in
(i) ,
(ii) ,
where denotes the indicator function of the set .
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4.II.25J
2005 commentLet be Borel-measurable. State Fubini's theorem for the double integral
Let . Show that the function
is measurable and integrable on .
Evaluate
by Fubini's theorem or otherwise.
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1.II.26I
2005 commentA cell has been placed in a biological solution at time . After an exponential time of rate , it is divided, producing cells with probability , with the mean value means that the cell dies . The same mechanism is applied to each of the living cells, independently.
(a) Let be the number of living cells in the solution by time . Prove that . [You may use without proof, if you wish, the fact that, if a positive function satisfies for and is differentiable at zero, then , for some
Let be the probability generating function of . Prove that it satisfies the following differential equation
(b) Now consider the case where each cell is divided in two cells . Let be the number of cells produced in the solution by time .
Calculate the distribution of . Is an inhomogeneous Poisson process? If so, what is its rate ? Justify your answer.
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2.II.26I
2005 commentWhat does it mean to say that is a renewal process?
Let be a renewal process with holding times and let . For , set . Show that
for all , with equality if .
Consider now the case where are exponential random variables. Show that
and that, as ,
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3.II.25I
2005 commentConsider an loss system with arrival rate and service-time distribution . Thus, arrivals form a Poisson process of rate , service times are independent with common distribution , there are servers and there is no space for waiting. Use Little's Lemma to obtain a relation between the long-run average occupancy and the stationary probability that the system is full.
Cafe-Bar Duo has 23 serving tables. Each table can be occupied either by one person or two. Customers arrive either singly or in a pair; if a table is empty they are seated and served immediately, otherwise, they leave. The times between arrivals are independent exponential random variables of mean . Each arrival is twice as likely to be a single person as a pair. A single customer stays for an exponential time of mean 20 , whereas a pair stays for an exponential time of mean 30 ; all these times are independent of each other and of the process of arrivals. The value of orders taken at each table is a constant multiple of the time that it is occupied.
Express the long-run rate of revenue of the cafe as a function of the probability that an arriving customer or pair of customers finds the cafe full.
By imagining a cafe with infinitely many tables, show that where is a Poisson random variable of parameter . Deduce that is very small. [Credit will be given for any useful numerical estimate, an upper bound of being sufficient for full credit.]
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4.II.26I
2005 commentA particle performs a continuous-time nearest neighbour random walk on a regular triangular lattice inside an angle , starting from the corner. See the diagram below. The jump rates are from the corner and in each of the six directions if the particle is inside the angle. However, if the particle is on the edge of the angle, the rate is along the edge away from the corner and to each of three other neighbouring sites in the angle. See the diagram below, where a typical trajectory is also shown.

The particle position at time is determined by its vertical level and its horizontal position . For , if then . Here are positions inside, and 0 and positions on the edge of the angle, at vertical level .
Let be the times of subsequent jumps of process and consider the embedded discrete-time Markov chains
where is the vertical level immediately after time is the horizontal position immediately after time , and is the horizontal position immediately before time . (a) Assume that is a Markov chain with transition probabilities
and that is a continuous-time Markov chain with rates
[You will be asked to justify these assumptions in part (b) of the question.] Determine whether the chains and are transient, positive recurrent or null recurrent.
(b) Now assume that, conditional on and previously passed vertical levels, the horizontal positions and are uniformly distributed on . In other words, for all attainable values and for all ,
Deduce that and are indeed Markov chains with transition probabilities and rates as in (a).
(c) Finally, prove property .
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1.II.27I
2005 commentState Wilks' Theorem on the asymptotic distribution of likelihood-ratio test statistics.
Suppose that are independent with common distribution, where the parameters and are both unknown. Find the likelihood-ratio test statistic for testing against unrestricted, and state its (approximate) distribution.
What is the form of the -test of against ? Explain why for large the likelihood-ratio test and the -test are nearly the same.
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2.II.27I
2005 comment(i) Suppose that is a multivariate normal vector with mean and covariance matrix , where and are both unknown, and denotes the identity matrix. Suppose that are linear subspaces of of dimensions and , where . Let denote orthogonal projection onto . Carefully derive the joint distribution of under the hypothesis . How could you use this to make a test of against ?
(ii) Suppose that students take exams, and that the mark of student in exam is modelled as
where , the are independent , and the parameters and are unknown. Construct a test of for all against .
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3.II.26I
2005 commentIn the context of decision theory, explain the meaning of the following italicized terms: loss function, decision rule, the risk of a decision rule, a Bayes rule with respect to prior , and an admissible rule. Explain how a Bayes rule with respect to a prior can be constructed.
Suppose that are independent with common distribution, where is supposed to have a prior density . In a decision-theoretic approach to estimating , we take a quadratic loss: . Write and .
By considering decision rules (estimators) of the form , prove that if then the estimator is not Bayes, for any choice of prior .
By considering decision rules of the form , prove that if then the estimator is not Bayes, for any choice of prior .
[You may use without proof the fact that, if has a distribution, then .]
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4.II.27I
2005 commentA group of hospitals is to be 'appraised'; the 'performance' of hospital has a prior distribution, different hospitals being independent. The 'performance' cannot be measured directly, so an expensive firm of management consultants has been hired to arrive at each hospital's Standardised Index of Quality [SIQ], this being a number for hospital related to by the commercially-sensitive formula
where the are independent with common distribution.
(i) Assume that and are known. What is the posterior distribution of given ? Suppose that hospital was the hospital with the lowest SIQ, with a value ; conditional on , what is the distribution of ?
(ii) Now, instead of assuming and known, suppose that has a Gamma prior with parameters , density
for known and , and that , where is a known constant. Find the posterior distribution of given . Comment briefly on the form of the distribution.
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1.II.28J
2005 commentLet be a zero-mean Gaussian vector, with covariance matrix . In a simple single-period economy with agents, agent will receive at time . If is a contingent claim to be paid at time 1 , define agent 's reservation bid price for , assuming his preferences are given by for any contingent claim .
Assuming that for each , where , show that agent 's reservation bid price for units of is
As , find the limit of agent 's per-unit reservation bid price for , and comment on the expression you obtain.
The agents bargain, and reach an equilibrium. Assuming that the contingent claim is in zero net supply, show that the equilibrium price of will be
where and . Show that at that price agent will choose to buy
units of .
By computing the improvement in agent 's expected utility, show that the value to agent of access to this market is equal to a fixed payment of
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2.II.28J
2005 comment(i) At the beginning of year , an investor makes decisions about his investment and consumption for the coming year. He first takes out an amount from his current wealth , and sets this aside for consumption. He splits his remaining wealth between a bank account (unit wealth invested at the start of the year will have grown to a sure amount by the end of the year), and the stock market. Unit wealth invested in the stock market will have become the random amount by the end of the year.
The investor's objective is to invest and consume so as to maximise the expected value of , where is strictly increasing and strictly convex. Consider the dynamic programming equation (Bellman equation) for his problem,
Explain all undefined notation, and explain briefly why the equation holds.
(ii) Supposing that the are independent and identically distributed, and that , where is different from 1 , find as explicitly as you can the form of the agent's optimal policy.
(iii) Return to the general problem of (i). Assuming that the sample space is finite, and that all suprema are attained, show that
where denotes the optimal consumption and wealth process for the problem. Explain the significance of each of these equalities.
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3.II.27J
2005 commentSuppose that over two periods a stock price moves on a binomial tree:

(a) Find an arbitrage opportunity when the riskless rate equals . Give precise details of when and how much you buy, borrow and sell.
(b) From here on, assume instead that the riskless rate equals . Determine the equivalent martingale measure. [No proof is required.]
(c) Determine the time-zero price of an American put with strike 15 and expiry 2 . Assume you sell it at this price. Which hedge do you put on at time zero? Consider the scenario of two bad periods. How does your hedge work?
(d) The buyer of the American put turns out to be an unsophisticated investor who fails to use his early exercise right when he should. Assume the first period was bad. How much profit can you make out of this? You should detail your exact strategy.
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4.II.28J
2005 comment(a) In the context of the Black-Scholes formula, let be spot price, be strike price, be time to maturity, and assume constant interest rate , volatility and absence of dividends. Write down explicitly the prices of a European call and put,
Use for the normal distribution function. [No proof is required.]
(b) From here on assume . Keeping fixed, we shorten the notation to and similarly for . Show that put-call symmetry holds:
Check homogeneity: for every real
(c) Show that the price of a down-and-out European call with strike and barrier is given by
(d)
(i) Specialize the last expression to and simplify.
(ii) Answer a popular interview question in investment banks: What is the fair value of a down-and-out call given that ? Identify the corresponding hedge. [It may be helpful to compute Delta first.]
(iii) Does this hedge work beyond the Black-Scholes model? When does it fail?
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2.II.29I
2005 commentExplain what is meant by a time-homogeneous discrete time Markov decision problem.
What is the positive programming case?
A discrete time Markov decision problem has state space . In state , two actions are possible. We may either stop and obtain a terminal reward , or may continue, in which case the subsequent state is equally likely to be or . In states 0 and stopping is automatic (with terminal rewards and respectively). Starting in state , denote by and the maximal expected terminal reward that can be obtained over the first steps and over the infinite horizon, respectively. Prove that .
Prove that is the smallest concave function such that for all .
Describe an optimal policy.
Suppose are distinct numbers. Show that the optimal policy is unique, or give a counter-example.
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3.II.28I
2005 commentConsider the problem
where for ,
is the control variable, and is Gaussian white noise. Show that the problem can be rewritten as one of controlling the scalar variable , where
By guessing the form of the optimal value function and ensuring it satisfies an appropriate optimality equation, show that the optimal control is
Is this certainty equivalence control?
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4.II.29I
2005 commentA continuous-time control problem is defined in terms of state variable and control . We desire to minimize , where is fixed and is unconstrained. Given and , describe further boundary conditions that can be used in conjunction with Pontryagin's maximum principle to find and the adjoint variables .
Company 1 wishes to steal customers from Company 2 and maximize the profit it obtains over an interval . Denoting by the number of customers of Company , and by the advertising effort of Company 1 , this leads to a problem
where , and is constrained to the interval . Assuming , use Pontryagin's maximum principle to show that the optimal advertising policy is bang-bang, and that there is just one change in advertising effort, at a time , where
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1.II.29C
2005 commentConsider the equation
where , to be solved for . State clearly what it means for a hypersurface
defined by a function , to be non-characteristic for . Does the non-characteristic condition hold when ?
Solve for with initial condition where . For the case discuss the limiting behaviour as .
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2.II.30C
2005 commentDefine a fundamental solution of a linear partial differential operator . Prove that the function
defines a distribution which is a fundamental solution of the operator given by
Hence find a solution to the equation
where for and for .
Consider the functional
Show that for all Schwartz functions that are not identically zero.
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3.II.29C
2005 commentWrite down a formula for the solution of the -dimensional heat equation
for a given Schwartz function; here and is taken in the variables . Show that
Consider the equation
where is a given Schwartz function. Show that has a solution of the form
where is a Schwartz function.
Prove that the solution of the initial value problem for with initial data satisfies
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4.II.30C
2005 commentWrite down the solution of the three-dimensional wave equation
for a Schwartz function . Here is taken in the variables and etc. State the "strong" form of Huygens principle for this solution. Using the method of descent, obtain the solution of the corresponding problem in two dimensions. State the "weak" form of Huygens principle for this solution.
Let be a solution of
Show that
where
Hence deduce, by integration of over the region
or otherwise, that satisfies the weak Huygens principle.
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1.II
2005 commentExplain what is meant by an asymptotic power series about for a real function of a real variable. Show that a convergent power series is also asymptotic.
Show further that an asymptotic power series is unique (assuming that it exists).
Let the function be defined for by
By suitably expanding the denominator of the integrand, or otherwise, show that, as ,
and that the error, when the series is stopped after terms, does not exceed the absolute value of the th term of the series.
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3.II
2005 commentExplain, without proof, how to obtain an asymptotic expansion, as , of
if it is known that possesses an asymptotic power series as .
Indicate the modification required to obtain an asymptotic expansion, under suitable conditions, of
Find an asymptotic expansion as of the function defined by
and its analytic continuation to . Where are the Stokes lines, that is, the critical lines separating the Stokes regions?
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4.II
2005 commentConsider the differential equation
where in an interval . Given a solution and a further smooth function , define
Show that, when is regarded as the independent variable, the function obeys the differential equation
where denotes .
Taking the choice
show that equation becomes
where
In the case that is negligible, deduce the Liouville-Green approximate solutions
Consider the Whittaker equation
where is a real constant. Show that the Liouville-Green approximation suggests the existence of solutions with asymptotic behaviour of the form
as .
Given that these asymptotic series may be differentiated term-by-term, show that
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1.II.31D
2005 commentLet satisfy the linear singular integral equation
where denotes the principal value integral and denotes a counterclockwise smooth closed contour, enclosing the origin but not the points .
(a) Formulate the associated Riemann-Hilbert problem.
(b) For this Riemann-Hilbert problem, find the index, the homogeneous canonical solution and the solvability condition.
(c) Find .
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2.II.31C
2005 commentSuppose satisfies the equation
where etc.
(a) Find the 1-soliton solution.
[You may use, without proof, the indefinite integral .]
(b) Express the self-similar solution of the equation in terms of a solution, denoted by , of the Painlevé II equation.
(c) Using the Ansatz
find a particular solution of the mKdV equation in terms of a solution of the Airy equation
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3.II.31A
2005 commentLet be an off-diagonal matrix. The matrix NLS equation
admits the Lax pair
where is a matrix and denotes the matrix commutator.
Let be a matrix-valued function decaying as . Let satisfy the -matrix Riemann-Hilbert problem
(a) Find expressions for and , in terms of the coefficients in the large expansion of , so that solves
and
(b) Use the result of (a) to establish that
(c) Show that the above results provide a linearization of the matrix NLS equation. What is the disadvantage of this approach in comparison with the inverse scattering method?
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1.II.32D
2005 commentA one-dimensional harmonic oscillator has Hamiltonian
where
Assuming the existence of a normalised state with , verify that
are eigenstates of with energies , to be determined, and that these states all have unit norm.
The Hamiltonian is now modified by a term
where is a positive integer. Use perturbation theory to find the change in the lowest energy level to order for any . [You may quote any standard formula you need.]
Compute by perturbation theory, again to order , the change in the first excited energy level when . Show that in this special case, , the exact change in all energy levels as a result of the perturbation is .
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2.II.32D
2005 commentThe components of are hermitian matrices obeying
for any unit vector . Show that these properties imply
for any constant vectors a and . Assuming that is real, explain why the matrix is unitary, and show that
Hence deduce that
where is any unit vector orthogonal to .
Write down an equation relating the matrices and the angular momentum operator for a particle of spin one half, and explain briefly the significance of the conditions . Show that if is a state with spin 'up' measured along the direction then, for a certain choice of is a state with spin 'up' measured along the direction .
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3.II.32D
2005 commentThe angular momentum operators and refer to independent systems, each with total angular momentum one. The combination of these systems has a basis of states which are of product form where and are the eigenvalues of and respectively. Let denote the alternative basis states which are simultaneous eigenstates of and , where is the combined angular momentum. What are the possible values of and ? Find expressions for all states with in terms of product states. How do these states behave when the constituent systems are interchanged?
Two spin-one particles and have no mutual interaction but they each move in a potential which is independent of spin. The single-particle energy levels and the corresponding wavefunctions are the same for either or . Given that , explain how to construct the two-particle states of lowest energy and combined total spin for the cases that (i) and are identical, and (ii) and are not identical.
[You may assume and use the result
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4.II.32D
2005 commentThe Hamiltonian for a quantum system in the Schrödinger picture is
where is independent of time and the parameter is small. Define the interaction picture corresponding to this Hamiltonian and derive a time evolution equation for interaction picture states.
Let and be eigenstates of with distinct eigenvalues and respectively. Show that if the system is initially in state then the probability of measuring it to be in state after a time is
Deduce that if , where is a time-independent operator and is a positive constant, then the probability for such a transition to have occurred after a very long time is approximately
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1.II.33B
2005 commentA beam of particles is incident on a central potential that vanishes for . Define the differential cross-section .
Given that each incoming particle has momentum , explain the relevance of solutions to the time-independent Schrödinger equation with the asymptotic form
as , where and . Write down a formula that determines in this case.
Write down the time-independent Schrödinger equation for a particle of mass and energy in a central potential , and show that it allows a solution of the form
Show that this is consistent with and deduce an expression for . Obtain the Born approximation for , and show that , where
Under what conditions is the Born approximation valid?
Obtain a formula for in terms of the scattering angle in the case that
for constants and . Hence show that is independent of in the limit , when expressed in terms of and the energy .
[You may assume that
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2.II.33B
2005 commentDescribe briefly the variational approach to the determination of an approximate ground state energy of a Hamiltonian .
Let and be two states, and consider the trial state
for real constants and . Given that
and that , obtain an upper bound on in terms of and .
The normalized ground-state wavefunction of the Hamiltonian
Verify that the ground state energy of is
Now consider the Hamiltonian
and let be its ground-state energy as a function of . Assuming that
use to compute and for and as given. Hence show that
Why should you expect this inequality to become an approximate equality for sufficiently large ? Describe briefly how this is relevant to molecular binding.
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3.II.33B
2005 commentLet be the set of lattice vectors of some lattice. Define the reciprocal lattice. What is meant by a Bravais lattice?
Let be mutually orthogonal unit vectors. A crystal has identical atoms at positions given by the vectors
where are arbitrary integers and is a constant. Show that these vectors define a Bravais lattice with basis vectors
Verify that a basis for the reciprocal lattice is
In Bragg scattering, an incoming plane wave of wave-vector is scattered to an outgoing wave of wave-vector . Explain why for some reciprocal lattice vector g. Given that is the scattering angle, show that
For the above lattice, explain why you would expect scattering through angles and such that
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4.II.33B
2005 commentA semiconductor has a valence energy band with energies and density of states , and a conduction energy band with energies and density of states . Assume that as , and that as . At zero temperature all states in the valence band are occupied and the conduction band is empty. Let be the number of holes in the valence band and the number of electrons in the conduction band at temperature . Under suitable approximations derive the result
where
Briefly describe how a semiconductor may conduct electricity but with a conductivity that is strongly temperature dependent.
Describe how doping of the semiconductor leads to . A junction is formed between an -type semiconductor, with donor atoms, and a -type semiconductor, with acceptor atoms. Show that there is a potential difference across the junction, where is the electron charge, and
Two semiconductors, one -type and one -type, are joined to make a closed circuit with two junctions. Explain why a current will flow around the circuit if the junctions are at different temperatures.
[The Fermi-Dirac distribution function at temperature and chemical potential is , where is the number of states with energy .
Note that .]
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2.II.34D
2005 commentWrite down the first law of thermodynamics in differential form applied to an infinitesimal reversible change.
Explain what is meant by an adiabatic change.
Starting with the first law in differential form, derive the Maxwell relation
Hence show that
For radiation in thermal equilibrium at temperature in volume , it is given that and . Hence deduce Stefan's Law,
where is a constant.
The radiation is allowed to expand adiabatically. Show that is constant during the expansion.
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3.II.34D
2005 commentA free spinless particle moving in two dimensions is confined to a square box of side . By imposing periodic boundary conditions show that the number of states in the energy range is , where
If, instead, the particle is an electron with magnetic moment moving in a constant external magnetic field , show that
Let there be electrons in the box. Explain briefly how to construct the ground state of the system. Let be the Fermi energy. Show that when
Show also that the magnetic moment of the system in its ground state is given by
and that the ground state energy is
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4.II.34D
2005 commentWrite down an expression for the partition function of a classical particle of mass moving in three dimensions in a potential and in equilibrium with a heat bath at temperature .
A system of non-interacting classical particles is placed in the potential
where is a positive integer. The gas is in equilibrium at temperature . Using a suitable rescaling of variables, show that the free energy is given by
where
Regarding as an external parameter, find the thermodynamic force , conjugate to , exerted by this system. Find the equation of state and compare with that of an ideal gas confined in a volume .
Derive expressions for the entropy , the internal energy and the total heat capacity at constant .
Show that for all the total heat capacity at constant is given by
[Note that
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1.II
2005 commentIn a frame the electromagnetic fields are encoded into the Maxwell field 4-tensor and its dual , where
and
[Here the signature is and units are chosen so that .] Obtain two independent Lorentz scalars of the electromagnetic field in terms of and .
Suppose that in the frame . Given that there exists a frame in which , show that
where are the magnitudes of , and
[Hint: there is no need to consider the Lorentz transformations for and .]
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3.II
2005 commentA non-relativistic particle of rest mass and charge is moving slowly with velocity . The power radiated per unit solid angle in the direction of a unit vector is
Obtain Larmor's formula
The particle has energy and, starting from afar, makes a head-on collision with a fixed central force described by a potential , where for and for . Let be the total energy radiated by the particle. Given that , show that
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4.II
2005 commentIn Ginzburg-Landau theory, superconductivity is due to "supercarriers" of mass and charge , which are described by a macroscopic wavefunction with "Mexican hat" potential
Here, is constant and is a function of temperature such that for but for , where is a critical temperature. In the presence of a magnetic field , the total energy of the superconducting system is
Use this to derive the equations
and
where
Suppose that we write the wavefunction as
where is the (real positive) supercarrier density and is a real phase function. Given that
show that is constant and that . Given also that , deduce that (*) allows such solutions for a certain choice of , which should be determined. Verify that your results imply . Show also that and hence that ( ) is solved.
Let be a surface within the superconductor with closed boundary . Show that the magnetic flux through is
Discuss, briefly, flux quantization.
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1.II.35C
2005 commentSuppose is a timelike geodesic of the metric
where is proper time along the world line. Show that , where is a constant whose physical significance should be stated. Setting , show that
Deduce that is a periodic function of proper time with period . Sketch as a function of and superpose on this a sketch of as a function of . Given the identity
deduce that is also a periodic function of with period .
Next consider the family of metrics
where is an odd function of , and for all . Derive expressions analogous to above. Deduce that is a periodic function of and also that is a periodic function of . What are the periods?
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2.II.35C
2005 commentState without proof the properties of local inertial coordinates centred on an arbitrary spacetime event . Explain their physical significance.
Obtain an expression for at in inertial coordinates. Use it to derive the formula
for the components of the Riemann tensor at in local inertial coordinates. Hence deduce that at any point in any chart .
Consider the metric
where is the Minkowski metric tensor and is a constant. Compute the Ricci scalar at the origin .
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4.II.36C
2005 commentState clearly, but do not prove, Birkhoff's Theorem about spherically symmetric spacetimes. Let be standard spherical polar coordinates and define , where is a constant. Consider the metric
Explain carefully why this is appropriate for the region outside a spherically symmetric star that is collapsing to form a black hole.
By considering radially infalling timelike geodesics , where is proper time along the curve, show that a freely falling observer will reach the event horizon after a finite proper time. Show also that a distant observer would see the horizon crossing only after an infinite time.
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1.II.36E
2005 commentConsider a unidirectional flow with dynamic viscosity along a straight rigid-walled channel of uniform cross-sectional shape driven by a uniform applied pressure gradient . Write down the differential equation and boundary conditions governing the velocity along the channel.
Consider the situation when the boundary includes a sharp corner of angle . Explain why one might expect that, sufficiently close to the corner, the solution should be of the form
where and are polar co-ordinates with origin at the vertex and describing the two planes emanating from the corner. Determine .
If is the sector bounded by the lines and the circular arc , show that the flow is given by
where and are to be determined.
[Note that .]
Considering the values of and , comment briefly on the cases: (i) ;
(ii) ; and (iii) .
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2.II.36E
2005 commentA volume of very viscous fluid of density and dynamic viscosity is released at the origin on a rigid horizontal boundary at time . Using lubrication theory, determine the velocity profile in the gravity current once it has spread sufficiently that the axisymmetric thickness of the current is much less than the radius of the front.
Derive the differential equation
where is to be determined.
Write down the other equations that are needed to determine the appropriate similarity solution for this problem.
Determine the similarity solution and calculate .
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2005
commentWrite down the Navier-Stokes equations for an incompressible fluid.
Explain the concepts of the Euler and Prandtl limits applied to the Navier-Stokes equations near a rigid boundary.
A steady two-dimensional flow given by far upstream flows past a semi-infinite flat plate, held at . Derive the boundary layer equation
for the stream-function near the plate, explaining any approximations made.
Show that the appropriate solution must be of the form
and determine the dimensionless variable .
Derive the equation and boundary conditions satisfied by . [You need not solve them.]
Suppose now that the plate has a finite length in the direction of the flow. Show that the force on the plate (per unit width perpendicular to the flow) is given by
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4.II.37E
2005 commentConsider flow of an incompressible fluid of uniform density and dynamic viscosity . Show that the rate of viscous dissipation per unit volume is given by
where is the strain rate.
Determine expressions for and when the flow is irrotational with velocity potential . Hence determine the rate of viscous dissipation, averaged over a wave period , for an irrotational two-dimensional surface wave of wavenumber and small amplitude in a fluid of very small viscosity and great depth .
[You may use without derivation that in deep water a linearised wave with surface displacement has velocity potential .]
Calculate the depth-integrated time-averaged kinetic energy per wavelength. Assuming that the average potential energy is equal to the average kinetic energy, show that the total wave energy decreases to leading order like , where
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1.II
2005 commentAn elastic solid with density has Lamé moduli and . Write down equations satisfied by the dilational and shear potentials and .
For a two-dimensional disturbance give expressions for the displacement field in terms of and .
Suppose the solid occupies the region and that the surface is free of traction. Find a combination of solutions for and that represent a propagating surface wave (a Rayleigh wave) near . Show that the wave is non-dispersive and obtain an equation for the speed . [You may assume without proof that this equation has a unique positive root.]
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2.II.37E
2005 commentShow that, in the standard notation for a one-dimensional flow of a perfect gas at constant entropy, the quantity remains constant along characteristics .
A perfect gas is initially at rest and occupies a tube in . A piston is pushed into the gas so that its position at time is , where is a constant. Find the time and position at which a shock first forms in the gas.
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3.II.37E
2005 commentThe real function satisfies the equation
where is a constant. Find the dispersion relation for waves of wavenumber and deduce whether wave crests move faster or slower than a wave packet.
Suppose that is given by a Fourier transform as
Use the method of stationary phase to find as for fixed .
[You may use the result that if
What can be said if ? [Detailed calculation is not required in this case.]
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4.II
2005 commentStarting from the equations of conservation of mass and momentum for an inviscid compressible fluid, show that for small perturbations about a state of rest and uniform density the velocity is irrotational and the velocity potential satisfies the wave equation. Identify the sound speed .
Define the acoustic energy density and acoustic energy flux, and derive the equation for conservation of acoustic energy.
Show that in any (not necessarily harmonic) acoustic plane wave of wavenumber the kinetic and potential energy densities are equal and that the acoustic energy is transported with velocity .
Calculate the kinetic and potential energy densities for a spherically symmetric outgoing wave. Are they equal?
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1.II.38A
2005 commentLet
where is a positive integer and ranges over all integers, be a finite-difference method for the advection equation
Here is the Courant number.
(a) Show that the local error of the method is .
(b) Determine the range of for which the method is stable.
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2.II.38A
2005 commentDefine a Krylov subspace .
Let be the dimension of . Prove that the sequence increases monotonically. Show that, moreover, there exists an integer with the following property: for , while for . Assuming that has a full set of eigenvectors, show that is equal to the number of eigenvectors of required to represent the vector .
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3.II.38A
2005 commentConsider the Runge-Kutta method
for the solution of the scalar ordinary differential equation . Here is a real parameter.
(a) Determine the order of the method.
(b) Find the range of values of for which the method is A-stable.
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4.II.39A
2005 commentAn skew-symmetric matrix is converted into an upper-Hessenberg form , say, by Householder reflections.
(a) Describe each step of the procedure and observe that is tridiagonal. Your algorithm should take advantage of the special form of to reduce the number of computations.
(b) Compare the cost (counting only products and looking only at the leading term) of converting a skew-symmetric and a symmetric matrix to an upper-Hessenberg form using Householder reflections.